8 3

8_2

8_4

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8 3 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X14,10,15,9 X10,5,11,6 X12,3,13,4 X4,11,5,12 X2,13,3,14 X16,8,1,7 X8,16,9,15
Gauss code	1, -6, 4, -5, 3, -1, 7, -8, 2, -3, 5, -4, 6, -2, 8, -7
Dowker-Thistlethwaite code	6 12 10 16 14 4 2 8
Conway Notation	[44]

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 2 3-genus 1 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-5] Hyperbolic Volume 5.23868 A-Polynomial See Data:8 3/A-polynomial

[edit Notes for 8 3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for 8 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-4t+9-4t^{-1}$
Conway polynomial	$1-4z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, 0 }
Jones polynomial	$q^{4}-q^{3}+2q^{2}-3q+3-3q^{-1}+2q^{-2}-q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}-z^{2}a^{2}-2z^{2}-1-z^{2}a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$az^{7}+z^{7}a^{-1}+a^{2}z^{6}+z^{6}a^{-2}+2z^{6}+a^{3}z^{5}-4az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}+a^{4}z^{4}-2a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-6z^{4}-2a^{3}z^{3}+8az^{3}+8z^{3}a^{-1}-2z^{3}a^{-3}-3a^{4}z^{2}+a^{2}z^{2}+z^{2}a^{-2}-3z^{2}a^{-4}+8z^{2}-4az-4za^{-1}+a^{4}+a^{-4}-1$
The A2 invariant	$q^{14}+q^{12}+q^{8}-q^{4}-1-q^{-4}+q^{-8}+q^{-12}+q^{-14}$
The G2 invariant	$q^{66}+q^{62}-q^{60}+q^{58}+q^{56}-q^{54}+2q^{52}-q^{50}+2q^{48}-q^{46}+q^{42}-2q^{40}+4q^{38}-3q^{36}+q^{34}+q^{32}-2q^{30}+3q^{28}-2q^{26}+q^{24}+2q^{22}-2q^{20}+q^{18}-2q^{14}+4q^{12}-4q^{10}+q^{8}-3q^{4}+3q^{2}-5+3q^{-2}-3q^{-4}+q^{-8}-4q^{-10}+4q^{-12}-2q^{-14}+q^{-18}-2q^{-20}+2q^{-22}+q^{-24}-2q^{-26}+3q^{-28}-2q^{-30}+q^{-32}+q^{-34}-3q^{-36}+4q^{-38}-2q^{-40}+q^{-42}-q^{-46}+2q^{-48}-q^{-50}+2q^{-52}-q^{-54}+q^{-56}+q^{-58}-q^{-60}+q^{-62}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 8_3.

A1 Invariants.

Weight	Invariant
1	$q^{9}+q^{5}-q^{3}-q^{-3}+q^{-5}+q^{-9}$
2	$q^{26}+q^{20}-q^{18}-2q^{16}+q^{14}-2q^{10}+2q^{8}+q^{6}-q^{4}+q^{2}+1+q^{-2}-q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-14}-2q^{-16}-q^{-18}+q^{-20}+q^{-26}$
3	$q^{51}-q^{41}-q^{39}+q^{35}-q^{33}-2q^{31}+3q^{27}+3q^{25}-2q^{23}-2q^{21}+2q^{19}+4q^{17}-2q^{15}-3q^{13}+q^{11}+3q^{9}-2q^{5}-2q^{-5}+3q^{-9}+q^{-11}-3q^{-13}-2q^{-15}+4q^{-17}+2q^{-19}-2q^{-21}-2q^{-23}+3q^{-25}+3q^{-27}-2q^{-31}-q^{-33}+q^{-35}-q^{-39}-q^{-41}+q^{-51}$
4	$q^{84}-q^{76}-q^{72}+q^{68}-q^{66}-2q^{62}-q^{60}+3q^{58}+2q^{56}+3q^{54}-2q^{52}-3q^{50}-q^{48}+q^{46}+7q^{44}+2q^{42}-3q^{40}-6q^{38}-5q^{36}+6q^{34}+6q^{32}+q^{30}-6q^{28}-9q^{26}+4q^{24}+6q^{22}+3q^{20}-3q^{18}-7q^{16}+q^{14}+3q^{12}+3q^{10}-2q^{6}+q^{4}+q^{2}+1+q^{-2}+q^{-4}-2q^{-6}+3q^{-10}+3q^{-12}+q^{-14}-7q^{-16}-3q^{-18}+3q^{-20}+6q^{-22}+4q^{-24}-9q^{-26}-6q^{-28}+q^{-30}+6q^{-32}+6q^{-34}-5q^{-36}-6q^{-38}-3q^{-40}+2q^{-42}+7q^{-44}+q^{-46}-q^{-48}-3q^{-50}-2q^{-52}+3q^{-54}+2q^{-56}+3q^{-58}-q^{-60}-2q^{-62}-q^{-66}+q^{-68}-q^{-72}-q^{-76}+q^{-84}$
5	$q^{125}-q^{117}-q^{115}+q^{107}-2q^{103}-q^{101}+q^{97}+3q^{95}+3q^{93}-3q^{89}-3q^{87}-2q^{85}+3q^{83}+5q^{81}+5q^{79}+q^{77}-5q^{75}-8q^{73}-6q^{71}+8q^{67}+10q^{65}+5q^{63}-6q^{61}-14q^{59}-11q^{57}+12q^{53}+16q^{51}+7q^{49}-11q^{47}-18q^{45}-10q^{43}+6q^{41}+18q^{39}+16q^{37}-3q^{35}-15q^{33}-12q^{31}+11q^{27}+12q^{25}+2q^{23}-7q^{21}-8q^{19}-2q^{17}+4q^{15}+4q^{13}+2q^{11}-q^{9}-3q^{7}-3q^{-7}-q^{-9}+2q^{-11}+4q^{-13}+4q^{-15}-2q^{-17}-8q^{-19}-7q^{-21}+2q^{-23}+12q^{-25}+11q^{-27}-12q^{-31}-15q^{-33}-3q^{-35}+16q^{-37}+18q^{-39}+6q^{-41}-10q^{-43}-18q^{-45}-11q^{-47}+7q^{-49}+16q^{-51}+12q^{-53}-11q^{-57}-14q^{-59}-6q^{-61}+5q^{-63}+10q^{-65}+8q^{-67}-6q^{-71}-8q^{-73}-5q^{-75}+q^{-77}+5q^{-79}+5q^{-81}+3q^{-83}-2q^{-85}-3q^{-87}-3q^{-89}+3q^{-93}+3q^{-95}+q^{-97}-q^{-101}-2q^{-103}+q^{-107}-q^{-115}-q^{-117}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$q^{14}+q^{12}+q^{8}-q^{4}-1-q^{-4}+q^{-8}+q^{-12}+q^{-14}$
1,1	$q^{36}+2q^{32}-2q^{30}+4q^{28}-2q^{26}+4q^{24}-6q^{22}+5q^{20}-6q^{18}+4q^{16}-6q^{14}-q^{12}-6q^{8}+8q^{6}-8q^{4}+16q^{2}-6+16q^{-2}-8q^{-4}+8q^{-6}-6q^{-8}-q^{-12}-6q^{-14}+4q^{-16}-6q^{-18}+5q^{-20}-6q^{-22}+4q^{-24}-2q^{-26}+4q^{-28}-2q^{-30}+2q^{-32}+q^{-36}$
2,0	$q^{36}+q^{34}+q^{32}+q^{28}+q^{26}-q^{24}-3q^{22}-2q^{20}-2q^{14}+2q^{10}+q^{4}+2q^{2}+2+2q^{-2}+q^{-4}+2q^{-10}-2q^{-14}-2q^{-20}-3q^{-22}-q^{-24}+q^{-26}+q^{-28}+q^{-32}+q^{-34}+q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{28}+q^{24}+q^{22}+q^{18}+2q^{16}-2q^{14}-q^{12}-3q^{8}-q^{6}+q^{4}+2q^{2}+2+2q^{-2}+q^{-4}-q^{-6}-3q^{-8}-q^{-12}-2q^{-14}+2q^{-16}+q^{-18}+q^{-22}+q^{-24}+q^{-28}$
1,0,0	$q^{19}+q^{17}+q^{15}+q^{11}-q^{5}-q-q^{-1}-q^{-5}+q^{-11}+q^{-15}+q^{-17}+q^{-19}$

B2 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{66}+q^{62}-q^{60}+q^{58}+q^{56}-q^{54}+2q^{52}-q^{50}+2q^{48}-q^{46}+q^{42}-2q^{40}+4q^{38}-3q^{36}+q^{34}+q^{32}-2q^{30}+3q^{28}-2q^{26}+q^{24}+2q^{22}-2q^{20}+q^{18}-2q^{14}+4q^{12}-4q^{10}+q^{8}-3q^{4}+3q^{2}-5+3q^{-2}-3q^{-4}+q^{-8}-4q^{-10}+4q^{-12}-2q^{-14}+q^{-18}-2q^{-20}+2q^{-22}+q^{-24}-2q^{-26}+3q^{-28}-2q^{-30}+q^{-32}+q^{-34}-3q^{-36}+4q^{-38}-2q^{-40}+q^{-42}-q^{-46}+2q^{-48}-q^{-50}+2q^{-52}-q^{-54}+q^{-56}+q^{-58}-q^{-60}+q^{-62}+q^{-66}$

.

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 3"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-4t+9-4t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$1-4z^{2}$

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]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{7}+z^{7}a^{-1}+a^{2}z^{6}+z^{6}a^{-2}+2z^{6}+a^{3}z^{5}-4az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}+a^{4}z^{4}-2a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-6z^{4}-2a^{3}z^{3}+8az^{3}+8z^{3}a^{-1}-2z^{3}a^{-3}-3a^{4}z^{2}+a^{2}z^{2}+z^{2}a^{-2}-3z^{2}a^{-4}+8z^{2}-4az-4za^{-1}+a^{4}+a^{-4}-1$

Vassiliev invariants

V2 and V3:

(-4, 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 $-16$ $0$ $128$ ${\frac {520}{3}}$ ${\frac {200}{3}}$ $0$ $0$ $0$ $0$ $-{\frac {2048}{3}}$ $0$ $-{\frac {8320}{3}}$ $-{\frac {3200}{3}}$ $-{\frac {37502}{15}}$ ${\frac {6728}{15}}$ $-{\frac {96968}{45}}$ ${\frac {2174}{9}}$ $-{\frac {7262}{15}}$

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 3. 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                   0 5             2 1   1 3           1       -1 1         2 2       0 -1       2 2         0 -3       1           -1 -5   1 2             1 -7                   0 -9 1                 1

Integral Khovanov Homology (db, data source)

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

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, 3]]
Out[2]=	8
In[3]:=	PD[Knot[8, 3]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 10, 15, 9], X[10, 5, 11, 6], X[12, 3, 13, 4], X[4, 11, 5, 12], X[2, 13, 3, 14], X[16, 8, 1, 7], X[8, 16, 9, 15]]
In[4]:=	GaussCode[Knot[8, 3]]
Out[4]=	GaussCode[1, -6, 4, -5, 3, -1, 7, -8, 2, -3, 5, -4, 6, -2, 8, -7]
In[5]:=	BR[Knot[8, 3]]
Out[5]=	BR[5, {-1, -1, -2, 1, 3, -2, 3, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[8, 3]][t]
Out[6]=	4 9 - - - 4 t t
In[7]:=	Conway[Knot[8, 3]][z]
Out[7]=	2 1 - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 3], Knot[10, 1]}
In[9]:=	{KnotDet[Knot[8, 3]], KnotSignature[Knot[8, 3]]}
Out[9]=	{17, 0}
In[10]:=	J=Jones[Knot[8, 3]][q]
Out[10]=	-4 -3 2 3 2 3 4 3 + q - q + -- - - - 3 q + 2 q - q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 3]}
In[12]:=	A2Invariant[Knot[8, 3]][q]
Out[12]=	-14 -12 -8 -4 4 8 12 14 -1 + q + q + q - q - q + q + q + q
In[13]:=	Kauffman[Knot[8, 3]][a, z]
Out[13]=	2 2 -4 4 4 z 2 3 z z 2 2 4 2 -1 + a + a - --- - 4 a z + 8 z - ---- + -- + a z - 3 a z - a 4 2 a a 3 3 4 4 2 z 8 z 3 3 3 4 z 2 z 2 4 4 4 ---- + ---- + 8 a z - 2 a z - 6 z + -- - ---- - 2 a z + a z + 3 a 4 2 a a a 5 5 6 7 z 4 z 5 3 5 6 z 2 6 z 7 -- - ---- - 4 a z + a z + 2 z + -- + a z + -- + a z 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[8, 3]], Vassiliev[3][Knot[8, 3]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[8, 3]][q, t]
Out[15]=	2 1 1 2 1 2 3 5 2 - + 2 q + ----- + ----- + ----- + ---- + --- + 2 q t + q t + 2 q t + q 9 4 5 3 5 2 3 q t q t q t q t q t 5 3 9 4 q t + q t