8 13

8_12

8_14

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8 13 Quick Notes

8 13 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X15,7,16,6 X7,15,8,14 X13,9,14,8 X9,2,10,3
Gauss code	-1, 8, -2, 1, -4, 5, -6, 7, -8, 2, -3, 4, -7, 6, -5, 3
Dowker-Thistlethwaite code	4 10 12 14 2 16 8 6
Conway Notation	[31112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-6] Hyperbolic Volume 8.53123 A-Polynomial See Data:8 13/A-polynomial

[edit Notes for 8 13'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 13's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-7t+11-7t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 29, 0 }
Jones polynomial	$-q^{5}+2q^{4}-3q^{3}+5q^{2}-5q+5-4q^{-1}+3q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+z^{2}+2a^{-2}-a^{-4}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-1}+z^{7}a^{-3}+5z^{6}a^{-2}+2z^{6}a^{-4}+3z^{6}+4az^{5}+4z^{5}a^{-1}+z^{5}a^{-3}+z^{5}a^{-5}+3a^{2}z^{4}-11z^{4}a^{-2}-6z^{4}a^{-4}-2z^{4}+a^{3}z^{3}-4az^{3}-9z^{3}a^{-1}-7z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+7z^{2}a^{-2}+5z^{2}a^{-4}+az+3za^{-1}+4za^{-3}+2za^{-5}-2a^{-2}-a^{-4}$
The A2 invariant	$-q^{10}+q^{8}+q^{6}-q^{4}+q^{2}-1+q^{-2}+q^{-4}+q^{-6}+2q^{-8}-q^{-10}-q^{-16}$
The G2 invariant	$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{44}-3q^{40}+9q^{38}-11q^{36}+12q^{34}-8q^{32}+6q^{28}-13q^{26}+18q^{24}-16q^{22}+10q^{20}-8q^{16}+14q^{14}-10q^{12}+4q^{10}+2q^{8}-10q^{6}+9q^{4}-3q^{2}-7+17q^{-2}-22q^{-4}+19q^{-6}-6q^{-8}-9q^{-10}+19q^{-12}-26q^{-14}+26q^{-16}-14q^{-18}+3q^{-20}+11q^{-22}-16q^{-24}+21q^{-26}-10q^{-28}+q^{-30}+6q^{-32}-9q^{-34}+8q^{-36}-6q^{-40}+14q^{-42}-15q^{-44}+9q^{-46}+q^{-48}-13q^{-50}+18q^{-52}-19q^{-54}+11q^{-56}-4q^{-58}-6q^{-60}+11q^{-62}-13q^{-64}+10q^{-66}-4q^{-68}-q^{-70}+2q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 8_13.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+2q^{5}-q^{3}+q+2q^{-5}-q^{-7}+q^{-9}-q^{-11}$
2	$q^{20}-2q^{18}-q^{16}+5q^{14}-4q^{12}-3q^{10}+7q^{8}-3q^{6}-3q^{4}+6q^{2}+1-2q^{-2}+3q^{-6}-q^{-8}-5q^{-10}+4q^{-12}+3q^{-14}-6q^{-16}+3q^{-18}+4q^{-20}-5q^{-22}+3q^{-26}-2q^{-28}-q^{-30}+q^{-32}$
3	$-q^{39}+2q^{37}+q^{35}-3q^{33}-2q^{31}+4q^{29}+6q^{27}-9q^{25}-8q^{23}+10q^{21}+11q^{19}-12q^{17}-15q^{15}+12q^{13}+19q^{11}-8q^{9}-18q^{7}+5q^{5}+16q^{3}+q-11q^{-1}-6q^{-3}+7q^{-5}+10q^{-7}-q^{-9}-12q^{-11}-3q^{-13}+14q^{-15}+8q^{-17}-15q^{-19}-11q^{-21}+13q^{-23}+13q^{-25}-10q^{-27}-17q^{-29}+8q^{-31}+17q^{-33}-2q^{-35}-17q^{-37}-q^{-39}+14q^{-41}+5q^{-43}-10q^{-45}-7q^{-47}+5q^{-49}+6q^{-51}-2q^{-53}-4q^{-55}+2q^{-59}+q^{-61}-q^{-63}$
4	$q^{64}-2q^{62}-q^{60}+3q^{58}+2q^{54}-7q^{52}-q^{50}+10q^{48}+q^{46}+3q^{44}-21q^{42}-6q^{40}+26q^{38}+13q^{36}-47q^{32}-19q^{30}+43q^{28}+40q^{26}+11q^{24}-67q^{22}-47q^{20}+36q^{18}+60q^{16}+35q^{14}-55q^{12}-60q^{10}+6q^{8}+46q^{6}+49q^{4}-18q^{2}-43-22q^{-2}+11q^{-4}+39q^{-6}+19q^{-8}-13q^{-10}-39q^{-12}-19q^{-14}+24q^{-16}+43q^{-18}+9q^{-20}-44q^{-22}-35q^{-24}+9q^{-26}+59q^{-28}+26q^{-30}-46q^{-32}-47q^{-34}-7q^{-36}+64q^{-38}+41q^{-40}-34q^{-42}-52q^{-44}-32q^{-46}+51q^{-48}+56q^{-50}-4q^{-52}-42q^{-54}-54q^{-56}+18q^{-58}+50q^{-60}+26q^{-62}-12q^{-64}-51q^{-66}-12q^{-68}+21q^{-70}+29q^{-72}+14q^{-74}-25q^{-76}-18q^{-78}-4q^{-80}+12q^{-82}+16q^{-84}-4q^{-86}-6q^{-88}-6q^{-90}+6q^{-94}+q^{-96}-2q^{-100}-q^{-102}+q^{-104}$
5	$-q^{95}+2q^{93}+q^{91}-3q^{89}+q^{83}+2q^{81}-6q^{77}-4q^{75}+7q^{73}+12q^{71}+4q^{69}-13q^{67}-20q^{65}-14q^{63}+20q^{61}+52q^{59}+24q^{57}-36q^{55}-78q^{53}-56q^{51}+38q^{49}+124q^{47}+100q^{45}-36q^{43}-161q^{41}-154q^{39}+5q^{37}+184q^{35}+211q^{33}+41q^{31}-183q^{29}-251q^{27}-94q^{25}+149q^{23}+266q^{21}+146q^{19}-95q^{17}-245q^{15}-175q^{13}+34q^{11}+195q^{9}+183q^{7}+27q^{5}-131q^{3}-166q-71q^{-1}+63q^{-3}+132q^{-5}+100q^{-7}+q^{-9}-98q^{-11}-115q^{-13}-48q^{-15}+65q^{-17}+128q^{-19}+82q^{-21}-39q^{-23}-136q^{-25}-111q^{-27}+27q^{-29}+148q^{-31}+130q^{-33}-18q^{-35}-157q^{-37}-154q^{-39}+6q^{-41}+171q^{-43}+177q^{-45}+11q^{-47}-176q^{-49}-200q^{-51}-41q^{-53}+166q^{-55}+225q^{-57}+77q^{-59}-142q^{-61}-231q^{-63}-119q^{-65}+90q^{-67}+226q^{-69}+164q^{-71}-37q^{-73}-191q^{-75}-185q^{-77}-36q^{-79}+139q^{-81}+190q^{-83}+86q^{-85}-73q^{-87}-162q^{-89}-121q^{-91}+8q^{-93}+117q^{-95}+127q^{-97}+42q^{-99}-62q^{-101}-107q^{-103}-66q^{-105}+13q^{-107}+70q^{-109}+69q^{-111}+18q^{-113}-35q^{-115}-53q^{-117}-29q^{-119}+7q^{-121}+30q^{-123}+28q^{-125}+6q^{-127}-14q^{-129}-17q^{-131}-7q^{-133}+2q^{-135}+8q^{-137}+8q^{-139}-4q^{-143}-3q^{-145}-q^{-147}+2q^{-151}+q^{-153}-q^{-155}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}+q^{8}+q^{6}-q^{4}+q^{2}-1+q^{-2}+q^{-4}+q^{-6}+2q^{-8}-q^{-10}-q^{-16}$
1,1	$q^{28}-4q^{26}+8q^{24}-12q^{22}+20q^{20}-32q^{18}+38q^{16}-44q^{14}+51q^{12}-52q^{10}+44q^{8}-30q^{6}+13q^{4}+12q^{2}-36+66q^{-2}-82q^{-4}+100q^{-6}-102q^{-8}+98q^{-10}-86q^{-12}+66q^{-14}-42q^{-16}+12q^{-18}+10q^{-20}-28q^{-22}+46q^{-24}-54q^{-26}+55q^{-28}-48q^{-30}+40q^{-32}-32q^{-34}+19q^{-36}-12q^{-38}+6q^{-40}-2q^{-42}+q^{-44}$
2,0	$q^{26}-q^{24}-2q^{22}+q^{20}+3q^{18}-5q^{14}+3q^{10}-2q^{8}-2q^{6}+4q^{4}+4q^{2}+1+q^{-2}+2q^{-4}-q^{-6}-2q^{-8}+q^{-10}-q^{-12}-3q^{-14}+3q^{-16}+4q^{-18}-q^{-20}-2q^{-22}+2q^{-24}+2q^{-26}-2q^{-28}-3q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-42}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{22}-2q^{20}-q^{18}+4q^{16}-3q^{14}-q^{12}+6q^{10}-3q^{8}-3q^{6}+4q^{4}-2q^{2}-2+2q^{-2}+2q^{-4}+q^{-6}+5q^{-10}+4q^{-12}-4q^{-14}+3q^{-16}+2q^{-18}-6q^{-20}+q^{-24}-4q^{-26}+q^{-28}+q^{-30}-q^{-32}+q^{-34}$
1,0,0	$-q^{13}+q^{11}+q^{7}-q^{5}+q^{3}-q+q^{-3}+q^{-5}+2q^{-7}+q^{-9}+2q^{-11}-q^{-13}-q^{-17}-q^{-21}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{28}-q^{26}-2q^{24}+q^{22}+2q^{20}-q^{18}-2q^{16}+3q^{14}+3q^{12}-3q^{10}-3q^{8}+4q^{6}-4q^{2}+1+2q^{-2}-2q^{-4}-q^{-6}+4q^{-8}+3q^{-10}+q^{-12}+7q^{-14}+8q^{-16}-q^{-20}+4q^{-22}-2q^{-24}-7q^{-26}-4q^{-28}-2q^{-32}-3q^{-34}+q^{-36}+2q^{-38}+q^{-44}$
1,0,0,0	$-q^{16}+q^{14}+q^{8}-q^{6}+q^{4}-q^{2}+q^{-4}+q^{-6}+2q^{-8}+2q^{-10}+q^{-12}+2q^{-14}-q^{-16}-q^{-20}-q^{-22}-q^{-26}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{22}+2q^{20}-3q^{18}+4q^{16}-5q^{14}+5q^{12}-4q^{10}+3q^{8}-q^{6}+4q^{2}-6+8q^{-2}-8q^{-4}+9q^{-6}-8q^{-8}+7q^{-10}-4q^{-12}+2q^{-14}+q^{-16}-2q^{-18}+4q^{-20}-4q^{-22}+5q^{-24}-4q^{-26}+3q^{-28}-3q^{-30}+q^{-32}-q^{-34}$
1,0	$q^{36}-2q^{32}-2q^{30}+q^{28}+4q^{26}+q^{24}-4q^{22}-3q^{20}+3q^{18}+5q^{16}-5q^{12}-2q^{10}+3q^{8}+3q^{6}-3q^{4}-3q^{2}+2+4q^{-2}-3q^{-6}+4q^{-10}+2q^{-12}-2q^{-14}-q^{-16}+4q^{-18}+3q^{-20}-2q^{-22}-4q^{-24}+2q^{-26}+5q^{-28}+q^{-30}-5q^{-32}-4q^{-34}+2q^{-36}+4q^{-38}-q^{-40}-4q^{-42}-2q^{-44}+2q^{-46}+2q^{-48}-q^{-50}-q^{-52}+q^{-56}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{30}-2q^{28}+q^{26}-2q^{24}+4q^{22}-4q^{20}+3q^{18}-3q^{16}+5q^{14}-2q^{12}+q^{10}-2q^{8}+2q^{4}-4q^{2}+3-6q^{-2}+7q^{-4}-5q^{-6}+8q^{-8}-5q^{-10}+9q^{-12}-q^{-14}+7q^{-16}-q^{-18}+2q^{-20}+q^{-22}-2q^{-24}-5q^{-28}+2q^{-30}-5q^{-32}+2q^{-34}-4q^{-36}+3q^{-38}-2q^{-40}+2q^{-42}-q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{44}-3q^{40}+9q^{38}-11q^{36}+12q^{34}-8q^{32}+6q^{28}-13q^{26}+18q^{24}-16q^{22}+10q^{20}-8q^{16}+14q^{14}-10q^{12}+4q^{10}+2q^{8}-10q^{6}+9q^{4}-3q^{2}-7+17q^{-2}-22q^{-4}+19q^{-6}-6q^{-8}-9q^{-10}+19q^{-12}-26q^{-14}+26q^{-16}-14q^{-18}+3q^{-20}+11q^{-22}-16q^{-24}+21q^{-26}-10q^{-28}+q^{-30}+6q^{-32}-9q^{-34}+8q^{-36}-6q^{-40}+14q^{-42}-15q^{-44}+9q^{-46}+q^{-48}-13q^{-50}+18q^{-52}-19q^{-54}+11q^{-56}-4q^{-58}-6q^{-60}+11q^{-62}-13q^{-64}+10q^{-66}-4q^{-68}-q^{-70}+2q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

.

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 13"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{2}-7t+11-7t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{-1}+z^{7}a^{-3}+5z^{6}a^{-2}+2z^{6}a^{-4}+3z^{6}+4az^{5}+4z^{5}a^{-1}+z^{5}a^{-3}+z^{5}a^{-5}+3a^{2}z^{4}-11z^{4}a^{-2}-6z^{4}a^{-4}-2z^{4}+a^{3}z^{3}-4az^{3}-9z^{3}a^{-1}-7z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+7z^{2}a^{-2}+5z^{2}a^{-4}+az+3za^{-1}+4za^{-3}+2za^{-5}-2a^{-2}-a^{-4}$

Vassiliev invariants

V2 and V3:

(1, 1)

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 $4$ $8$ $8$ ${\frac {14}{3}}$ $-{\frac {38}{3}}$ $32$ ${\frac {80}{3}}$ ${\frac {32}{3}}$ $-24$ ${\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ $-{\frac {152}{3}}$ ${\frac {1951}{30}}$ ${\frac {526}{5}}$ $-{\frac {6058}{45}}$ $-{\frac {31}{18}}$ $-{\frac {1409}{30}}$

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 13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-3	-2	-1	0	1	2	3	4	5	χ
11									1	-1
9								1		1
7							2	1		-1
5						3	1			2
3					2	2				0
1				3	3					0
-1			2	3						1
-3		1	2							-1
-5		2								2
-7	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, 13]]
Out[2]=	8
In[3]:=	PD[Knot[8, 13]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 16], X[5, 13, 6, 12], X[15, 7, 16, 6], X[7, 15, 8, 14], X[13, 9, 14, 8], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[8, 13]]
Out[4]=	GaussCode[-1, 8, -2, 1, -4, 5, -6, 7, -8, 2, -3, 4, -7, 6, -5, 3]
In[5]:=	BR[Knot[8, 13]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, 2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[8, 13]][t]
Out[6]=	2 7 2 11 + -- - - - 7 t + 2 t 2 t t
In[7]:=	Conway[Knot[8, 13]][z]
Out[7]=	2 4 1 + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 13]}
In[9]:=	{KnotDet[Knot[8, 13]], KnotSignature[Knot[8, 13]]}
Out[9]=	{29, 0}
In[10]:=	J=Jones[Knot[8, 13]][q]
Out[10]=	-3 3 4 2 3 4 5 5 - q + -- - - - 5 q + 5 q - 3 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 13]}
In[12]:=	A2Invariant[Knot[8, 13]][q]
Out[12]=	-10 -8 -6 -4 -2 2 4 6 8 10 16 -1 - q + q + q - q + q + q + q + q + 2 q - q - q
In[13]:=	Kauffman[Knot[8, 13]][a, z]
Out[13]=	2 2 3 -4 2 2 z 4 z 3 z 5 z 7 z 2 2 3 z -a - -- + --- + --- + --- + a z + ---- + ---- - 2 a z - ---- - 2 5 3 a 4 2 5 a a a a a a 3 3 4 4 5 7 z 9 z 3 3 3 4 6 z 11 z 2 4 z ---- - ---- - 4 a z + a z - 2 z - ---- - ----- + 3 a z + -- + 3 a 4 2 5 a a a a 5 5 6 6 7 7 z 4 z 5 6 2 z 5 z z z -- + ---- + 4 a z + 3 z + ---- + ---- + -- + -- 3 a 4 2 3 a a a a a
In[14]:=	{Vassiliev[2][Knot[8, 13]], Vassiliev[3][Knot[8, 13]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[8, 13]][q, t]
Out[15]=	3 1 2 1 2 2 3 - + 3 q + ----- + ----- + ----- + ---- + --- + 3 q t + 2 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 2 q t + 3 q t + q t + 2 q t + q t + q t + q t