8 12

8_11

8_13

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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 $\{4,6\}$ 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 $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

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

Polynomial invariants

Alexander polynomial	$t^{2}-7t+13-7t^{-1}+t^{-2}$
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]=

$t^{2}-7t+13-7t^{-1}+t^{-2}$

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 $-12$ $0$ $72$ $82$ $30$ $0$ $0$ $0$ $0$ $-288$ $0$ $-984$ $-360$ $-{\frac {8031}{10}}$ ${\frac {1226}{15}}$ $-{\frac {9262}{15}}$ ${\frac {479}{6}}$ $-{\frac {1311}{10}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{12}-2q^{11}+6q^{9}-8q^{8}-3q^{7}+18q^{6}-15q^{5}-10q^{4}+30q^{3}-18q^{2}-16q+35-16q^{-1}-18q^{-2}+30q^{-3}-10q^{-4}-15q^{-5}+18q^{-6}-3q^{-7}-8q^{-8}+6q^{-9}-2q^{-11}+q^{-12}$
3	$q^{24}-2q^{23}+2q^{21}+3q^{20}-7q^{19}-5q^{18}+11q^{17}+13q^{16}-18q^{15}-22q^{14}+20q^{13}+40q^{12}-26q^{11}-54q^{10}+23q^{9}+73q^{8}-20q^{7}-88q^{6}+15q^{5}+100q^{4}-9q^{3}-108q^{2}+4q+109+4q^{-1}-108q^{-2}-9q^{-3}+100q^{-4}+15q^{-5}-88q^{-6}-20q^{-7}+73q^{-8}+23q^{-9}-54q^{-10}-26q^{-11}+40q^{-12}+20q^{-13}-22q^{-14}-18q^{-15}+13q^{-16}+11q^{-17}-5q^{-18}-7q^{-19}+3q^{-20}+2q^{-21}-2q^{-23}+q^{-24}$
4	$q^{40}-2q^{39}+2q^{37}-q^{36}+4q^{35}-9q^{34}-q^{33}+10q^{32}+q^{31}+13q^{30}-32q^{29}-14q^{28}+25q^{27}+19q^{26}+46q^{25}-72q^{24}-58q^{23}+23q^{22}+54q^{21}+130q^{20}-105q^{19}-134q^{18}-21q^{17}+81q^{16}+255q^{15}-101q^{14}-209q^{13}-104q^{12}+77q^{11}+381q^{10}-64q^{9}-257q^{8}-187q^{7}+52q^{6}+464q^{5}-20q^{4}-267q^{3}-244q^{2}+18q+493+18q^{-1}-244q^{-2}-267q^{-3}-20q^{-4}+464q^{-5}+52q^{-6}-187q^{-7}-257q^{-8}-64q^{-9}+381q^{-10}+77q^{-11}-104q^{-12}-209q^{-13}-101q^{-14}+255q^{-15}+81q^{-16}-21q^{-17}-134q^{-18}-105q^{-19}+130q^{-20}+54q^{-21}+23q^{-22}-58q^{-23}-72q^{-24}+46q^{-25}+19q^{-26}+25q^{-27}-14q^{-28}-32q^{-29}+13q^{-30}+q^{-31}+10q^{-32}-q^{-33}-9q^{-34}+4q^{-35}-q^{-36}+2q^{-37}-2q^{-39}+q^{-40}$
5	$q^{60}-2q^{59}+2q^{57}-q^{56}+2q^{54}-5q^{53}-2q^{52}+9q^{51}+3q^{50}-2q^{49}-3q^{48}-18q^{47}-8q^{46}+22q^{45}+32q^{44}+14q^{43}-20q^{42}-63q^{41}-52q^{40}+30q^{39}+99q^{38}+101q^{37}-q^{36}-149q^{35}-185q^{34}-39q^{33}+177q^{32}+285q^{31}+144q^{30}-202q^{29}-411q^{28}-251q^{27}+169q^{26}+520q^{25}+428q^{24}-118q^{23}-632q^{22}-588q^{21}+23q^{20}+695q^{19}+777q^{18}+91q^{17}-748q^{16}-931q^{15}-217q^{14}+766q^{13}+1064q^{12}+339q^{11}-761q^{10}-1171q^{9}-444q^{8}+743q^{7}+1238q^{6}+535q^{5}-705q^{4}-1286q^{3}-605q^{2}+666q+1291+666q^{-1}-605q^{-2}-1286q^{-3}-705q^{-4}+535q^{-5}+1238q^{-6}+743q^{-7}-444q^{-8}-1171q^{-9}-761q^{-10}+339q^{-11}+1064q^{-12}+766q^{-13}-217q^{-14}-931q^{-15}-748q^{-16}+91q^{-17}+777q^{-18}+695q^{-19}+23q^{-20}-588q^{-21}-632q^{-22}-118q^{-23}+428q^{-24}+520q^{-25}+169q^{-26}-251q^{-27}-411q^{-28}-202q^{-29}+144q^{-30}+285q^{-31}+177q^{-32}-39q^{-33}-185q^{-34}-149q^{-35}-q^{-36}+101q^{-37}+99q^{-38}+30q^{-39}-52q^{-40}-63q^{-41}-20q^{-42}+14q^{-43}+32q^{-44}+22q^{-45}-8q^{-46}-18q^{-47}-3q^{-48}-2q^{-49}+3q^{-50}+9q^{-51}-2q^{-52}-5q^{-53}+2q^{-54}-q^{-56}+2q^{-57}-2q^{-59}+q^{-60}$
6	$q^{84}-2q^{83}+2q^{81}-q^{80}-2q^{78}+6q^{77}-6q^{76}-3q^{75}+11q^{74}-q^{73}-2q^{72}-12q^{71}+11q^{70}-16q^{69}-7q^{68}+38q^{67}+16q^{66}+3q^{65}-41q^{64}+q^{63}-68q^{62}-33q^{61}+98q^{60}+93q^{59}+77q^{58}-58q^{57}-37q^{56}-237q^{55}-183q^{54}+124q^{53}+255q^{52}+331q^{51}+93q^{50}+5q^{49}-556q^{48}-604q^{47}-102q^{46}+356q^{45}+782q^{44}+595q^{43}+402q^{42}-827q^{41}-1294q^{40}-786q^{39}+85q^{38}+1183q^{37}+1419q^{36}+1336q^{35}-712q^{34}-1957q^{33}-1851q^{32}-715q^{31}+1194q^{30}+2244q^{29}+2655q^{28}-93q^{27}-2262q^{26}-2933q^{25}-1845q^{24}+746q^{23}+2752q^{22}+3943q^{21}+806q^{20}-2153q^{19}-3708q^{18}-2911q^{17}+70q^{16}+2891q^{15}+4876q^{14}+1634q^{13}-1812q^{12}-4100q^{11}-3656q^{10}-556q^{9}+2791q^{8}+5381q^{7}+2212q^{6}-1428q^{5}-4191q^{4}-4055q^{3}-1036q^{2}+2568q+5533+2568q^{-1}-1036q^{-2}-4055q^{-3}-4191q^{-4}-1428q^{-5}+2212q^{-6}+5381q^{-7}+2791q^{-8}-556q^{-9}-3656q^{-10}-4100q^{-11}-1812q^{-12}+1634q^{-13}+4876q^{-14}+2891q^{-15}+70q^{-16}-2911q^{-17}-3708q^{-18}-2153q^{-19}+806q^{-20}+3943q^{-21}+2752q^{-22}+746q^{-23}-1845q^{-24}-2933q^{-25}-2262q^{-26}-93q^{-27}+2655q^{-28}+2244q^{-29}+1194q^{-30}-715q^{-31}-1851q^{-32}-1957q^{-33}-712q^{-34}+1336q^{-35}+1419q^{-36}+1183q^{-37}+85q^{-38}-786q^{-39}-1294q^{-40}-827q^{-41}+402q^{-42}+595q^{-43}+782q^{-44}+356q^{-45}-102q^{-46}-604q^{-47}-556q^{-48}+5q^{-49}+93q^{-50}+331q^{-51}+255q^{-52}+124q^{-53}-183q^{-54}-237q^{-55}-37q^{-56}-58q^{-57}+77q^{-58}+93q^{-59}+98q^{-60}-33q^{-61}-68q^{-62}+q^{-63}-41q^{-64}+3q^{-65}+16q^{-66}+38q^{-67}-7q^{-68}-16q^{-69}+11q^{-70}-12q^{-71}-2q^{-72}-q^{-73}+11q^{-74}-3q^{-75}-6q^{-76}+6q^{-77}-2q^{-78}-q^{-80}+2q^{-81}-2q^{-83}+q^{-84}$
7	$q^{112}-2q^{111}+2q^{109}-q^{108}-2q^{106}+2q^{105}+5q^{104}-7q^{103}-q^{102}+7q^{101}-q^{100}-12q^{98}-2q^{97}+17q^{96}-13q^{95}+3q^{94}+22q^{93}+7q^{92}+7q^{91}-43q^{90}-35q^{89}+10q^{88}-26q^{87}+25q^{86}+81q^{85}+63q^{84}+69q^{83}-77q^{82}-148q^{81}-104q^{80}-156q^{79}+17q^{78}+206q^{77}+282q^{76}+356q^{75}+55q^{74}-268q^{73}-435q^{72}-661q^{71}-340q^{70}+204q^{69}+654q^{68}+1130q^{67}+788q^{66}+49q^{65}-750q^{64}-1693q^{63}-1545q^{62}-598q^{61}+680q^{60}+2252q^{59}+2499q^{58}+1547q^{57}-198q^{56}-2717q^{55}-3672q^{54}-2860q^{53}-655q^{52}+2844q^{51}+4768q^{50}+4537q^{49}+2104q^{48}-2562q^{47}-5828q^{46}-6373q^{45}-3891q^{44}+1784q^{43}+6441q^{42}+8212q^{41}+6132q^{40}-511q^{39}-6764q^{38}-9928q^{37}-8380q^{36}-1108q^{35}+6552q^{34}+11294q^{33}+10685q^{32}+2988q^{31}-6052q^{30}-12328q^{29}-12727q^{28}-4870q^{27}+5238q^{26}+12969q^{25}+14476q^{24}+6663q^{23}-4286q^{22}-13308q^{21}-15867q^{20}-8241q^{19}+3344q^{18}+13388q^{17}+16876q^{16}+9543q^{15}-2421q^{14}-13285q^{13}-17618q^{12}-10578q^{11}+1640q^{10}+13082q^{9}+18052q^{8}+11361q^{7}-915q^{6}-12789q^{5}-18322q^{4}-11965q^{3}+311q^{2}+12426q+18379+12426q^{-1}+311q^{-2}-11965q^{-3}-18322q^{-4}-12789q^{-5}-915q^{-6}+11361q^{-7}+18052q^{-8}+13082q^{-9}+1640q^{-10}-10578q^{-11}-17618q^{-12}-13285q^{-13}-2421q^{-14}+9543q^{-15}+16876q^{-16}+13388q^{-17}+3344q^{-18}-8241q^{-19}-15867q^{-20}-13308q^{-21}-4286q^{-22}+6663q^{-23}+14476q^{-24}+12969q^{-25}+5238q^{-26}-4870q^{-27}-12727q^{-28}-12328q^{-29}-6052q^{-30}+2988q^{-31}+10685q^{-32}+11294q^{-33}+6552q^{-34}-1108q^{-35}-8380q^{-36}-9928q^{-37}-6764q^{-38}-511q^{-39}+6132q^{-40}+8212q^{-41}+6441q^{-42}+1784q^{-43}-3891q^{-44}-6373q^{-45}-5828q^{-46}-2562q^{-47}+2104q^{-48}+4537q^{-49}+4768q^{-50}+2844q^{-51}-655q^{-52}-2860q^{-53}-3672q^{-54}-2717q^{-55}-198q^{-56}+1547q^{-57}+2499q^{-58}+2252q^{-59}+680q^{-60}-598q^{-61}-1545q^{-62}-1693q^{-63}-750q^{-64}+49q^{-65}+788q^{-66}+1130q^{-67}+654q^{-68}+204q^{-69}-340q^{-70}-661q^{-71}-435q^{-72}-268q^{-73}+55q^{-74}+356q^{-75}+282q^{-76}+206q^{-77}+17q^{-78}-156q^{-79}-104q^{-80}-148q^{-81}-77q^{-82}+69q^{-83}+63q^{-84}+81q^{-85}+25q^{-86}-26q^{-87}+10q^{-88}-35q^{-89}-43q^{-90}+7q^{-91}+7q^{-92}+22q^{-93}+3q^{-94}-13q^{-95}+17q^{-96}-2q^{-97}-12q^{-98}-q^{-100}+7q^{-101}-q^{-102}-7q^{-103}+5q^{-104}+2q^{-105}-2q^{-106}-q^{-108}+2q^{-109}-2q^{-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.

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