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]

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]=

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

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

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[8, 12]]
Out[2]=	8
In[3]:=	PD[Knot[8, 12]]
Out[3]=	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[4]:=	GaussCode[Knot[8, 12]]
Out[4]=	GaussCode[1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6]
In[5]:=	BR[Knot[8, 12]]
Out[5]=	BR[5, {-1, 2, -1, -3, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[8, 12]][t]
Out[6]=	-2 7 2 13 + t - - - 7 t + t t
In[7]:=	Conway[Knot[8, 12]][z]
Out[7]=	2 4 1 - 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 12]}
In[9]:=	{KnotDet[Knot[8, 12]], KnotSignature[Knot[8, 12]]}
Out[9]=	{29, 0}
In[10]:=	J=Jones[Knot[8, 12]][q]
Out[10]=	-4 2 4 5 2 3 4 5 + q - -- + -- - - - 5 q + 4 q - 2 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 12]}
In[12]:=	A2Invariant[Knot[8, 12]][q]
Out[12]=	-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[13]:=	Kauffman[Knot[8, 12]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[8, 12]], Vassiliev[3][Knot[8, 12]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[8, 12]][q, t]
Out[15]=	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