8 12

8_11

8_13

Visit 8 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 12's page at Knotilus! Visit 8 12's page at the original Knot Atlas! 8 12 Quick Notes

In symmetric decorative form

Knot presentations

Planar diagram presentation	X4251 X10,8,11,7 X8394 X2,9,3,10 X14,6,15,5 X16,11,1,12 X12,15,13,16 X6,14,7,13
Gauss code	1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6
Dowker-Thistlethwaite code	4 8 14 10 2 16 6 12
Conway Notation	[2222]

Minimum Braid Representative: Length is 8, width is 5. Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,6\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-5] Hyperbolic Volume 8.93586 A-Polynomial See Data:8 12/A-polynomial

[edit Notes for 8 12's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Rasmussen s-Invariant 0

[edit Notes for 8 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2-7 t+13-7 t^{-1} + t^{-2} }
Conway polynomial	$z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 29, 0 }
Jones polynomial	$q^{4}-2q^{3}+4q^{2}-5q+5-5q^{-1}+4q^{-2}-2q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}-2z^{2}a^{2}-a^{2}+z^{4}+z^{2}+1-2z^{2}a^{-2}-a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$az^{7}+z^{7}a^{-1}+2a^{2}z^{6}+2z^{6}a^{-2}+4z^{6}+2a^{3}z^{5}+2az^{5}+2z^{5}a^{-1}+2z^{5}a^{-3}+a^{4}z^{4}-a^{2}z^{4}-z^{4}a^{-2}+z^{4}a^{-4}-4z^{4}-3a^{3}z^{3}-3az^{3}-3z^{3}a^{-1}-3z^{3}a^{-3}-2a^{4}z^{2}-2a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}a^{-4}+a^{3}z+za^{-3}+a^{4}+a^{2}+a^{-2}+a^{-4}+1$
The A2 invariant	$q^{14}+q^{12}-q^{10}+q^{8}-q^{4}+q^{2}-1+q^{-2}-q^{-4}+q^{-8}-q^{-10}+q^{-12}+q^{-14}$
The G2 invariant	$q^{66}-q^{64}+3q^{62}-3q^{60}+2q^{58}-3q^{54}+9q^{52}-11q^{50}+12q^{48}-8q^{46}+10q^{42}-17q^{40}+23q^{38}-18q^{36}+8q^{34}+4q^{32}-16q^{30}+17q^{28}-13q^{26}+3q^{24}+6q^{22}-12q^{20}+9q^{18}-12q^{14}+21q^{12}-23q^{10}+15q^{8}-q^{6}-14q^{4}+27q^{2}-29+27q^{-2}-14q^{-4}-q^{-6}+15q^{-8}-23q^{-10}+21q^{-12}-12q^{-14}+9q^{-18}-12q^{-20}+6q^{-22}+3q^{-24}-13q^{-26}+17q^{-28}-16q^{-30}+4q^{-32}+8q^{-34}-18q^{-36}+23q^{-38}-17q^{-40}+10q^{-42}-8q^{-46}+12q^{-48}-11q^{-50}+9q^{-52}-3q^{-54}+2q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 8_12.

A1 Invariants.

Weight	Invariant
1	$q^{9}-q^{7}+2q^{5}-q^{3}-q^{-3}+2q^{-5}-q^{-7}+q^{-9}$
2	$q^{26}-q^{24}-q^{22}+4q^{20}-2q^{18}-5q^{16}+7q^{14}-7q^{10}+5q^{8}+2q^{6}-4q^{4}+q^{2}+3+q^{-2}-4q^{-4}+2q^{-6}+5q^{-8}-7q^{-10}+7q^{-14}-5q^{-16}-2q^{-18}+4q^{-20}-q^{-22}-q^{-24}+q^{-26}$
3	$q^{51}-q^{49}-q^{47}+q^{45}+3q^{43}-2q^{41}-7q^{39}+2q^{37}+12q^{35}+q^{33}-16q^{31}-7q^{29}+20q^{27}+12q^{25}-20q^{23}-17q^{21}+16q^{19}+22q^{17}-12q^{15}-20q^{13}+7q^{11}+18q^{9}-2q^{7}-13q^{5}-4q^{3}+9q+9q^{-1}-4q^{-3}-13q^{-5}-2q^{-7}+18q^{-9}+7q^{-11}-20q^{-13}-12q^{-15}+22q^{-17}+16q^{-19}-17q^{-21}-20q^{-23}+12q^{-25}+20q^{-27}-7q^{-29}-16q^{-31}+q^{-33}+12q^{-35}+2q^{-37}-7q^{-39}-2q^{-41}+3q^{-43}+q^{-45}-q^{-47}-q^{-49}+q^{-51}$
4	$q^{84}-q^{82}-q^{80}+q^{78}+3q^{74}-4q^{72}-5q^{70}+3q^{68}+5q^{66}+14q^{64}-9q^{62}-22q^{60}-7q^{58}+11q^{56}+44q^{54}+4q^{52}-40q^{50}-42q^{48}-7q^{46}+77q^{44}+44q^{42}-32q^{40}-76q^{38}-49q^{36}+76q^{34}+80q^{32}+5q^{30}-78q^{28}-82q^{26}+44q^{24}+81q^{22}+33q^{20}-50q^{18}-75q^{16}+8q^{14}+52q^{12}+42q^{10}-15q^{8}-49q^{6}-20q^{4}+18q^{2}+41+18q^{-2}-20q^{-4}-49q^{-6}-15q^{-8}+42q^{-10}+52q^{-12}+8q^{-14}-75q^{-16}-50q^{-18}+33q^{-20}+81q^{-22}+44q^{-24}-82q^{-26}-78q^{-28}+5q^{-30}+80q^{-32}+76q^{-34}-49q^{-36}-76q^{-38}-32q^{-40}+44q^{-42}+77q^{-44}-7q^{-46}-42q^{-48}-40q^{-50}+4q^{-52}+44q^{-54}+11q^{-56}-7q^{-58}-22q^{-60}-9q^{-62}+14q^{-64}+5q^{-66}+3q^{-68}-5q^{-70}-4q^{-72}+3q^{-74}+q^{-78}-q^{-80}-q^{-82}+q^{-84}$
5	$q^{125}-q^{123}-q^{121}+q^{119}+q^{113}-2q^{111}-4q^{109}+3q^{107}+7q^{105}+5q^{103}-13q^{99}-19q^{97}-6q^{95}+23q^{93}+39q^{91}+22q^{89}-23q^{87}-67q^{85}-59q^{83}+8q^{81}+95q^{79}+114q^{77}+28q^{75}-105q^{73}-174q^{71}-96q^{69}+88q^{67}+233q^{65}+180q^{63}-46q^{61}-258q^{59}-266q^{57}-31q^{55}+253q^{53}+337q^{51}+116q^{49}-221q^{47}-367q^{45}-192q^{43}+157q^{41}+366q^{39}+250q^{37}-93q^{35}-333q^{33}-262q^{31}+25q^{29}+273q^{27}+260q^{25}+20q^{23}-207q^{21}-230q^{19}-56q^{17}+140q^{15}+196q^{13}+81q^{11}-80q^{9}-157q^{7}-104q^{5}+27q^{3}+127q+127q^{-1}+27q^{-3}-104q^{-5}-157q^{-7}-80q^{-9}+81q^{-11}+196q^{-13}+140q^{-15}-56q^{-17}-230q^{-19}-207q^{-21}+20q^{-23}+260q^{-25}+273q^{-27}+25q^{-29}-262q^{-31}-333q^{-33}-93q^{-35}+250q^{-37}+366q^{-39}+157q^{-41}-192q^{-43}-367q^{-45}-221q^{-47}+116q^{-49}+337q^{-51}+253q^{-53}-31q^{-55}-266q^{-57}-258q^{-59}-46q^{-61}+180q^{-63}+233q^{-65}+88q^{-67}-96q^{-69}-174q^{-71}-105q^{-73}+28q^{-75}+114q^{-77}+95q^{-79}+8q^{-81}-59q^{-83}-67q^{-85}-23q^{-87}+22q^{-89}+39q^{-91}+23q^{-93}-6q^{-95}-19q^{-97}-13q^{-99}+5q^{-103}+7q^{-105}+3q^{-107}-4q^{-109}-2q^{-111}+q^{-113}+q^{-119}-q^{-121}-q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$q^{14}+q^{12}-q^{10}+q^{8}-q^{4}+q^{2}-1+q^{-2}-q^{-4}+q^{-8}-q^{-10}+q^{-12}+q^{-14}$
1,1	$q^{36}-2q^{34}+6q^{32}-10q^{30}+19q^{28}-30q^{26}+42q^{24}-54q^{22}+64q^{20}-70q^{18}+62q^{16}-46q^{14}+23q^{12}+10q^{10}-44q^{8}+80q^{6}-106q^{4}+124q^{2}-130+124q^{-2}-106q^{-4}+80q^{-6}-44q^{-8}+10q^{-10}+23q^{-12}-46q^{-14}+62q^{-16}-70q^{-18}+64q^{-20}-54q^{-22}+42q^{-24}-30q^{-26}+19q^{-28}-10q^{-30}+6q^{-32}-2q^{-34}+q^{-36}$
2,0	$q^{36}+q^{34}-2q^{30}+3q^{26}-4q^{22}-q^{20}+4q^{18}+2q^{16}-5q^{14}+4q^{10}-q^{8}-2q^{6}+q^{4}+2q^{2}+2q^{-2}+q^{-4}-2q^{-6}-q^{-8}+4q^{-10}-5q^{-14}+2q^{-16}+4q^{-18}-q^{-20}-4q^{-22}+3q^{-26}-2q^{-30}+q^{-34}+q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{28}-q^{26}+q^{24}+3q^{22}-3q^{20}+q^{18}+5q^{16}-6q^{14}+4q^{10}-5q^{8}-q^{6}+3q^{4}+q^{2}+q^{-2}+3q^{-4}-q^{-6}-5q^{-8}+4q^{-10}-6q^{-14}+5q^{-16}+q^{-18}-3q^{-20}+3q^{-22}+q^{-24}-q^{-26}+q^{-28}$
1,0,0	$q^{19}+q^{17}+q^{15}-q^{13}+q^{11}-q^{9}-q^{5}+q^{3}+q^{-3}-q^{-5}-q^{-9}+q^{-11}-q^{-13}+q^{-15}+q^{-17}+q^{-19}$

B2 Invariants.

Weight	Invariant
0,1	$q^{28}-q^{26}+3q^{24}-3q^{22}+5q^{20}-5q^{18}+5q^{16}-4q^{14}+2q^{12}-3q^{8}+5q^{6}-7q^{4}+9q^{2}-10+9q^{-2}-7q^{-4}+5q^{-6}-3q^{-8}+2q^{-12}-4q^{-14}+5q^{-16}-5q^{-18}+5q^{-20}-3q^{-22}+3q^{-24}-q^{-26}+q^{-28}$
1,0	$q^{46}-q^{42}-q^{40}+2q^{38}+3q^{36}-4q^{32}-2q^{30}+4q^{28}+5q^{26}-2q^{24}-6q^{22}-q^{20}+5q^{18}+2q^{16}-4q^{14}-3q^{12}+2q^{10}+3q^{8}-q^{6}-3q^{4}+q^{2}+5+q^{-2}-3q^{-4}-q^{-6}+3q^{-8}+2q^{-10}-3q^{-12}-4q^{-14}+2q^{-16}+5q^{-18}-q^{-20}-6q^{-22}-2q^{-24}+5q^{-26}+4q^{-28}-2q^{-30}-4q^{-32}+3q^{-36}+2q^{-38}-q^{-40}-q^{-42}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{66}-q^{64}+3q^{62}-3q^{60}+2q^{58}-3q^{54}+9q^{52}-11q^{50}+12q^{48}-8q^{46}+10q^{42}-17q^{40}+23q^{38}-18q^{36}+8q^{34}+4q^{32}-16q^{30}+17q^{28}-13q^{26}+3q^{24}+6q^{22}-12q^{20}+9q^{18}-12q^{14}+21q^{12}-23q^{10}+15q^{8}-q^{6}-14q^{4}+27q^{2}-29+27q^{-2}-14q^{-4}-q^{-6}+15q^{-8}-23q^{-10}+21q^{-12}-12q^{-14}+9q^{-18}-12q^{-20}+6q^{-22}+3q^{-24}-13q^{-26}+17q^{-28}-16q^{-30}+4q^{-32}+8q^{-34}-18q^{-36}+23q^{-38}-17q^{-40}+10q^{-42}-8q^{-46}+12q^{-48}-11q^{-50}+9q^{-52}-3q^{-54}+2q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 12"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2-7 t+13-7 t^{-1} + t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}-3z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 29, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{4}-2q^{3}+4q^{2}-5q+5-5q^{-1}+4q^{-2}-2q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$a^{4}-2z^{2}a^{2}-a^{2}+z^{4}+z^{2}+1-2z^{2}a^{-2}-a^{-2}+a^{-4}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$az^{7}+z^{7}a^{-1}+2a^{2}z^{6}+2z^{6}a^{-2}+4z^{6}+2a^{3}z^{5}+2az^{5}+2z^{5}a^{-1}+2z^{5}a^{-3}+a^{4}z^{4}-a^{2}z^{4}-z^{4}a^{-2}+z^{4}a^{-4}-4z^{4}-3a^{3}z^{3}-3az^{3}-3z^{3}a^{-1}-3z^{3}a^{-3}-2a^{4}z^{2}-2a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}a^{-4}+a^{3}z+za^{-3}+a^{4}+a^{2}+a^{-2}+a^{-4}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {...}

Vassiliev invariants

V2 and V3:

(-3, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-12$ $0$ $72$ $82$ $30$ $0$ $0$ $0$ $0$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -288} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -984} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -360} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{8031}{10}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1226}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{9262}{15}} ${\frac {479}{6}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1311}{10}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of 8 12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 9                 1 1 7               1   -1 5             3 1   2 3           2 1     -1 1         3 3       0 -1       3 3         0 -3     1 2           -1 -5   1 3             2 -7   1               -1 -9 1                 1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-2 q^{11}+6 q^9-8 q^8-3 q^7+18 q^6-15 q^5-10 q^4+30 q^3-18 q^2-16 q+35-16 q^{-1} -18 q^{-2} +30 q^{-3} -10 q^{-4} -15 q^{-5} +18 q^{-6} -3 q^{-7} -8 q^{-8} +6 q^{-9} -2 q^{-11} + q^{-12} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{24}-2 q^{23}+2 q^{21}+3 q^{20}-7 q^{19}-5 q^{18}+11 q^{17}+13 q^{16}-18 q^{15}-22 q^{14}+20 q^{13}+40 q^{12}-26 q^{11}-54 q^{10}+23 q^9+73 q^8-20 q^7-88 q^6+15 q^5+100 q^4-9 q^3-108 q^2+4 q+109+4 q^{-1} -108 q^{-2} -9 q^{-3} +100 q^{-4} +15 q^{-5} -88 q^{-6} -20 q^{-7} +73 q^{-8} +23 q^{-9} -54 q^{-10} -26 q^{-11} +40 q^{-12} +20 q^{-13} -22 q^{-14} -18 q^{-15} +13 q^{-16} +11 q^{-17} -5 q^{-18} -7 q^{-19} +3 q^{-20} +2 q^{-21} -2 q^{-23} + q^{-24} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{40}-2 q^{39}+2 q^{37}-q^{36}+4 q^{35}-9 q^{34}-q^{33}+10 q^{32}+q^{31}+13 q^{30}-32 q^{29}-14 q^{28}+25 q^{27}+19 q^{26}+46 q^{25}-72 q^{24}-58 q^{23}+23 q^{22}+54 q^{21}+130 q^{20}-105 q^{19}-134 q^{18}-21 q^{17}+81 q^{16}+255 q^{15}-101 q^{14}-209 q^{13}-104 q^{12}+77 q^{11}+381 q^{10}-64 q^9-257 q^8-187 q^7+52 q^6+464 q^5-20 q^4-267 q^3-244 q^2+18 q+493+18 q^{-1} -244 q^{-2} -267 q^{-3} -20 q^{-4} +464 q^{-5} +52 q^{-6} -187 q^{-7} -257 q^{-8} -64 q^{-9} +381 q^{-10} +77 q^{-11} -104 q^{-12} -209 q^{-13} -101 q^{-14} +255 q^{-15} +81 q^{-16} -21 q^{-17} -134 q^{-18} -105 q^{-19} +130 q^{-20} +54 q^{-21} +23 q^{-22} -58 q^{-23} -72 q^{-24} +46 q^{-25} +19 q^{-26} +25 q^{-27} -14 q^{-28} -32 q^{-29} +13 q^{-30} + q^{-31} +10 q^{-32} - q^{-33} -9 q^{-34} +4 q^{-35} - q^{-36} +2 q^{-37} -2 q^{-39} + q^{-40} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-2 q^{59}+2 q^{57}-q^{56}+2 q^{54}-5 q^{53}-2 q^{52}+9 q^{51}+3 q^{50}-2 q^{49}-3 q^{48}-18 q^{47}-8 q^{46}+22 q^{45}+32 q^{44}+14 q^{43}-20 q^{42}-63 q^{41}-52 q^{40}+30 q^{39}+99 q^{38}+101 q^{37}-q^{36}-149 q^{35}-185 q^{34}-39 q^{33}+177 q^{32}+285 q^{31}+144 q^{30}-202 q^{29}-411 q^{28}-251 q^{27}+169 q^{26}+520 q^{25}+428 q^{24}-118 q^{23}-632 q^{22}-588 q^{21}+23 q^{20}+695 q^{19}+777 q^{18}+91 q^{17}-748 q^{16}-931 q^{15}-217 q^{14}+766 q^{13}+1064 q^{12}+339 q^{11}-761 q^{10}-1171 q^9-444 q^8+743 q^7+1238 q^6+535 q^5-705 q^4-1286 q^3-605 q^2+666 q+1291+666 q^{-1} -605 q^{-2} -1286 q^{-3} -705 q^{-4} +535 q^{-5} +1238 q^{-6} +743 q^{-7} -444 q^{-8} -1171 q^{-9} -761 q^{-10} +339 q^{-11} +1064 q^{-12} +766 q^{-13} -217 q^{-14} -931 q^{-15} -748 q^{-16} +91 q^{-17} +777 q^{-18} +695 q^{-19} +23 q^{-20} -588 q^{-21} -632 q^{-22} -118 q^{-23} +428 q^{-24} +520 q^{-25} +169 q^{-26} -251 q^{-27} -411 q^{-28} -202 q^{-29} +144 q^{-30} +285 q^{-31} +177 q^{-32} -39 q^{-33} -185 q^{-34} -149 q^{-35} - q^{-36} +101 q^{-37} +99 q^{-38} +30 q^{-39} -52 q^{-40} -63 q^{-41} -20 q^{-42} +14 q^{-43} +32 q^{-44} +22 q^{-45} -8 q^{-46} -18 q^{-47} -3 q^{-48} -2 q^{-49} +3 q^{-50} +9 q^{-51} -2 q^{-52} -5 q^{-53} +2 q^{-54} - q^{-56} +2 q^{-57} -2 q^{-59} + q^{-60} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{84}-2 q^{83}+2 q^{81}-q^{80}-2 q^{78}+6 q^{77}-6 q^{76}-3 q^{75}+11 q^{74}-q^{73}-2 q^{72}-12 q^{71}+11 q^{70}-16 q^{69}-7 q^{68}+38 q^{67}+16 q^{66}+3 q^{65}-41 q^{64}+q^{63}-68 q^{62}-33 q^{61}+98 q^{60}+93 q^{59}+77 q^{58}-58 q^{57}-37 q^{56}-237 q^{55}-183 q^{54}+124 q^{53}+255 q^{52}+331 q^{51}+93 q^{50}+5 q^{49}-556 q^{48}-604 q^{47}-102 q^{46}+356 q^{45}+782 q^{44}+595 q^{43}+402 q^{42}-827 q^{41}-1294 q^{40}-786 q^{39}+85 q^{38}+1183 q^{37}+1419 q^{36}+1336 q^{35}-712 q^{34}-1957 q^{33}-1851 q^{32}-715 q^{31}+1194 q^{30}+2244 q^{29}+2655 q^{28}-93 q^{27}-2262 q^{26}-2933 q^{25}-1845 q^{24}+746 q^{23}+2752 q^{22}+3943 q^{21}+806 q^{20}-2153 q^{19}-3708 q^{18}-2911 q^{17}+70 q^{16}+2891 q^{15}+4876 q^{14}+1634 q^{13}-1812 q^{12}-4100 q^{11}-3656 q^{10}-556 q^9+2791 q^8+5381 q^7+2212 q^6-1428 q^5-4191 q^4-4055 q^3-1036 q^2+2568 q+5533+2568 q^{-1} -1036 q^{-2} -4055 q^{-3} -4191 q^{-4} -1428 q^{-5} +2212 q^{-6} +5381 q^{-7} +2791 q^{-8} -556 q^{-9} -3656 q^{-10} -4100 q^{-11} -1812 q^{-12} +1634 q^{-13} +4876 q^{-14} +2891 q^{-15} +70 q^{-16} -2911 q^{-17} -3708 q^{-18} -2153 q^{-19} +806 q^{-20} +3943 q^{-21} +2752 q^{-22} +746 q^{-23} -1845 q^{-24} -2933 q^{-25} -2262 q^{-26} -93 q^{-27} +2655 q^{-28} +2244 q^{-29} +1194 q^{-30} -715 q^{-31} -1851 q^{-32} -1957 q^{-33} -712 q^{-34} +1336 q^{-35} +1419 q^{-36} +1183 q^{-37} +85 q^{-38} -786 q^{-39} -1294 q^{-40} -827 q^{-41} +402 q^{-42} +595 q^{-43} +782 q^{-44} +356 q^{-45} -102 q^{-46} -604 q^{-47} -556 q^{-48} +5 q^{-49} +93 q^{-50} +331 q^{-51} +255 q^{-52} +124 q^{-53} -183 q^{-54} -237 q^{-55} -37 q^{-56} -58 q^{-57} +77 q^{-58} +93 q^{-59} +98 q^{-60} -33 q^{-61} -68 q^{-62} + q^{-63} -41 q^{-64} +3 q^{-65} +16 q^{-66} +38 q^{-67} -7 q^{-68} -16 q^{-69} +11 q^{-70} -12 q^{-71} -2 q^{-72} - q^{-73} +11 q^{-74} -3 q^{-75} -6 q^{-76} +6 q^{-77} -2 q^{-78} - q^{-80} +2 q^{-81} -2 q^{-83} + q^{-84} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{112}-2 q^{111}+2 q^{109}-q^{108}-2 q^{106}+2 q^{105}+5 q^{104}-7 q^{103}-q^{102}+7 q^{101}-q^{100}-12 q^{98}-2 q^{97}+17 q^{96}-13 q^{95}+3 q^{94}+22 q^{93}+7 q^{92}+7 q^{91}-43 q^{90}-35 q^{89}+10 q^{88}-26 q^{87}+25 q^{86}+81 q^{85}+63 q^{84}+69 q^{83}-77 q^{82}-148 q^{81}-104 q^{80}-156 q^{79}+17 q^{78}+206 q^{77}+282 q^{76}+356 q^{75}+55 q^{74}-268 q^{73}-435 q^{72}-661 q^{71}-340 q^{70}+204 q^{69}+654 q^{68}+1130 q^{67}+788 q^{66}+49 q^{65}-750 q^{64}-1693 q^{63}-1545 q^{62}-598 q^{61}+680 q^{60}+2252 q^{59}+2499 q^{58}+1547 q^{57}-198 q^{56}-2717 q^{55}-3672 q^{54}-2860 q^{53}-655 q^{52}+2844 q^{51}+4768 q^{50}+4537 q^{49}+2104 q^{48}-2562 q^{47}-5828 q^{46}-6373 q^{45}-3891 q^{44}+1784 q^{43}+6441 q^{42}+8212 q^{41}+6132 q^{40}-511 q^{39}-6764 q^{38}-9928 q^{37}-8380 q^{36}-1108 q^{35}+6552 q^{34}+11294 q^{33}+10685 q^{32}+2988 q^{31}-6052 q^{30}-12328 q^{29}-12727 q^{28}-4870 q^{27}+5238 q^{26}+12969 q^{25}+14476 q^{24}+6663 q^{23}-4286 q^{22}-13308 q^{21}-15867 q^{20}-8241 q^{19}+3344 q^{18}+13388 q^{17}+16876 q^{16}+9543 q^{15}-2421 q^{14}-13285 q^{13}-17618 q^{12}-10578 q^{11}+1640 q^{10}+13082 q^9+18052 q^8+11361 q^7-915 q^6-12789 q^5-18322 q^4-11965 q^3+311 q^2+12426 q+18379+12426 q^{-1} +311 q^{-2} -11965 q^{-3} -18322 q^{-4} -12789 q^{-5} -915 q^{-6} +11361 q^{-7} +18052 q^{-8} +13082 q^{-9} +1640 q^{-10} -10578 q^{-11} -17618 q^{-12} -13285 q^{-13} -2421 q^{-14} +9543 q^{-15} +16876 q^{-16} +13388 q^{-17} +3344 q^{-18} -8241 q^{-19} -15867 q^{-20} -13308 q^{-21} -4286 q^{-22} +6663 q^{-23} +14476 q^{-24} +12969 q^{-25} +5238 q^{-26} -4870 q^{-27} -12727 q^{-28} -12328 q^{-29} -6052 q^{-30} +2988 q^{-31} +10685 q^{-32} +11294 q^{-33} +6552 q^{-34} -1108 q^{-35} -8380 q^{-36} -9928 q^{-37} -6764 q^{-38} -511 q^{-39} +6132 q^{-40} +8212 q^{-41} +6441 q^{-42} +1784 q^{-43} -3891 q^{-44} -6373 q^{-45} -5828 q^{-46} -2562 q^{-47} +2104 q^{-48} +4537 q^{-49} +4768 q^{-50} +2844 q^{-51} -655 q^{-52} -2860 q^{-53} -3672 q^{-54} -2717 q^{-55} -198 q^{-56} +1547 q^{-57} +2499 q^{-58} +2252 q^{-59} +680 q^{-60} -598 q^{-61} -1545 q^{-62} -1693 q^{-63} -750 q^{-64} +49 q^{-65} +788 q^{-66} +1130 q^{-67} +654 q^{-68} +204 q^{-69} -340 q^{-70} -661 q^{-71} -435 q^{-72} -268 q^{-73} +55 q^{-74} +356 q^{-75} +282 q^{-76} +206 q^{-77} +17 q^{-78} -156 q^{-79} -104 q^{-80} -148 q^{-81} -77 q^{-82} +69 q^{-83} +63 q^{-84} +81 q^{-85} +25 q^{-86} -26 q^{-87} +10 q^{-88} -35 q^{-89} -43 q^{-90} +7 q^{-91} +7 q^{-92} +22 q^{-93} +3 q^{-94} -13 q^{-95} +17 q^{-96} -2 q^{-97} -12 q^{-98} - q^{-100} +7 q^{-101} - q^{-102} -7 q^{-103} +5 q^{-104} +2 q^{-105} -2 q^{-106} - q^{-108} +2 q^{-109} -2 q^{-111} + q^{-112} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[8, 12]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 8, 11, 7], X[8, 3, 9, 4], X[2, 9, 3, 10], X[14, 6, 15, 5], X[16, 11, 1, 12], X[12, 15, 13, 16], X[6, 14, 7, 13]]
In[3]:=	GaussCode[Knot[8, 12]]
Out[3]=	GaussCode[1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6]
In[4]:=	DTCode[Knot[8, 12]]
Out[4]=	DTCode[4, 8, 14, 10, 2, 16, 6, 12]
In[5]:=	br = BR[Knot[8, 12]]
Out[5]=	BR[5, {-1, 2, -1, -3, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 8}
In[7]:=	BraidIndex[Knot[8, 12]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[8, 12]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[8, 12]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{FullyAmphicheiral, 2, 2, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[8, 12]][t]
Out[10]=	-2 7 2 13 + t - - - 7 t + t t
In[11]:=	Conway[Knot[8, 12]][z]
Out[11]=	2 4 1 - 3 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 12]}
In[13]:=	{KnotDet[Knot[8, 12]], KnotSignature[Knot[8, 12]]}
Out[13]=	{29, 0}
In[14]:=	Jones[Knot[8, 12]][q]
Out[14]=	-4 2 4 5 2 3 4 5 + q - -- + -- - - - 5 q + 4 q - 2 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 12]}
In[16]:=	A2Invariant[Knot[8, 12]][q]
Out[16]=	-14 -12 -10 -8 -4 -2 2 4 8 10 12 -1 + q + q - q + q - q + q + q - q + q - q + q + 14 q
In[17]:=	HOMFLYPT[Knot[8, 12]][a, z]
Out[17]=	2 -4 -2 2 4 2 2 z 2 2 4 1 + a - a - a + a + z - ---- - 2 a z + z 2 a
In[18]:=	Kauffman[Knot[8, 12]][a, z]
Out[18]=	2 2 -4 -2 2 4 z 3 2 z 2 z 2 2 4 2 1 + a + a + a + a + -- + a z - ---- - ---- - 2 a z - 2 a z - 3 4 2 a a a 3 3 4 4 3 z 3 z 3 3 3 4 z z 2 4 4 4 ---- - ---- - 3 a z - 3 a z - 4 z + -- - -- - a z + a z + 3 a 4 2 a a a 5 5 6 7 2 z 2 z 5 3 5 6 2 z 2 6 z 7 ---- + ---- + 2 a z + 2 a z + 4 z + ---- + 2 a z + -- + a z 3 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[8, 12]], Vassiliev[3][Knot[8, 12]]}
Out[19]=	{-3, 0}
In[20]:=	Kh[Knot[8, 12]][q, t]
Out[20]=	3 1 1 1 3 1 2 3 - + 3 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 2 q t + q t + 3 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[8, 12], 2][q]
Out[21]=	-12 2 6 8 3 18 15 10 30 18 16 35 + q - --- + -- - -- - -- + -- - -- - -- + -- - -- - -- - 16 q - 11 9 8 7 6 5 4 3 2 q q q q q q q q q q 2 3 4 5 6 7 8 9 11 18 q + 30 q - 10 q - 15 q + 18 q - 3 q - 8 q + 6 q - 2 q + 12 q

See/edit the Rolfsen_Splice_Template. Back to the top.

8_11 8_13