K11a81

K11a80

K11a82

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	K11a81 Quick Notes

K11a81 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X22,10,1,9 X2,11,3,12 X18,13,19,14 X20,16,21,15 X8,17,9,18 X6,20,7,19 X16,22,17,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -5
Dowker-Thistlethwaite code	4 10 12 14 22 2 18 20 8 6 16
Conway Notation	[.212.20]

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a81/ThurstonBennequinNumber Hyperbolic Volume 16.051 A-Polynomial See Data:K11a81/A-polynomial

[edit Notes for K11a81's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -2

[edit Notes for K11a81's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 127, 2 }
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-13q^{4}+18q^{3}-20q^{2}+20q-17+13q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}+6z^{2}+1$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+5az^{9}+12z^{9}a^{-1}+7z^{9}a^{-3}+4a^{2}z^{8}+15z^{8}a^{-2}+11z^{8}a^{-4}+8z^{8}+a^{3}z^{7}-13az^{7}-26z^{7}a^{-1}-z^{7}a^{-3}+11z^{7}a^{-5}-14a^{2}z^{6}-50z^{6}a^{-2}-14z^{6}a^{-4}+8z^{6}a^{-6}-42z^{6}-3a^{3}z^{5}+3az^{5}-23z^{5}a^{-3}-13z^{5}a^{-5}+4z^{5}a^{-7}+15a^{2}z^{4}+39z^{4}a^{-2}+2z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+44z^{4}+3a^{3}z^{3}+9az^{3}+18z^{3}a^{-1}+19z^{3}a^{-3}+5z^{3}a^{-5}-2z^{3}a^{-7}-5a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}a^{-6}-13z^{2}-a^{3}z-4az-6za^{-1}-4za^{-3}-za^{-5}+1$
The A2 invariant	$-q^{12}+q^{10}+q^{8}-q^{6}+3q^{4}-3q^{2}+1+q^{-2}-2q^{-4}+5q^{-6}-3q^{-8}+3q^{-10}-q^{-12}-2q^{-14}+2q^{-16}-2q^{-18}+q^{-20}$
The G2 invariant	$q^{60}-3q^{58}+9q^{56}-19q^{54}+28q^{52}-32q^{50}+15q^{48}+29q^{46}-92q^{44}+157q^{42}-186q^{40}+140q^{38}-15q^{36}-174q^{34}+360q^{32}-449q^{30}+392q^{28}-169q^{26}-152q^{24}+452q^{22}-607q^{20}+545q^{18}-277q^{16}-87q^{14}+395q^{12}-526q^{10}+425q^{8}-144q^{6}-183q^{4}+416q^{2}-444+250q^{-2}+81q^{-4}-420q^{-6}+618q^{-8}-586q^{-10}+326q^{-12}+84q^{-14}-495q^{-16}+758q^{-18}-764q^{-20}+514q^{-22}-89q^{-24}-345q^{-26}+628q^{-28}-662q^{-30}+448q^{-32}-89q^{-34}-249q^{-36}+431q^{-38}-389q^{-40}+158q^{-42}+136q^{-44}-356q^{-46}+405q^{-48}-270q^{-50}+10q^{-52}+254q^{-54}-425q^{-56}+448q^{-58}-321q^{-60}+111q^{-62}+110q^{-64}-276q^{-66}+338q^{-68}-310q^{-70}+215q^{-72}-84q^{-74}-35q^{-76}+125q^{-78}-169q^{-80}+165q^{-82}-127q^{-84}+72q^{-86}-16q^{-88}-29q^{-90}+53q^{-92}-62q^{-94}+51q^{-96}-31q^{-98}+15q^{-100}+q^{-102}-8q^{-104}+10q^{-106}-10q^{-108}+6q^{-110}-3q^{-112}+q^{-114}$

Further Quantum Invariants

K11a81 Quantum Invariants.

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["K11a81"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-2z^{6}+1$

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]=

{ 127, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-4q^{6}+8q^{5}-13q^{4}+18q^{3}-20q^{2}+20q-17+13q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}+6z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+5az^{9}+12z^{9}a^{-1}+7z^{9}a^{-3}+4a^{2}z^{8}+15z^{8}a^{-2}+11z^{8}a^{-4}+8z^{8}+a^{3}z^{7}-13az^{7}-26z^{7}a^{-1}-z^{7}a^{-3}+11z^{7}a^{-5}-14a^{2}z^{6}-50z^{6}a^{-2}-14z^{6}a^{-4}+8z^{6}a^{-6}-42z^{6}-3a^{3}z^{5}+3az^{5}-23z^{5}a^{-3}-13z^{5}a^{-5}+4z^{5}a^{-7}+15a^{2}z^{4}+39z^{4}a^{-2}+2z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+44z^{4}+3a^{3}z^{3}+9az^{3}+18z^{3}a^{-1}+19z^{3}a^{-3}+5z^{3}a^{-5}-2z^{3}a^{-7}-5a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}a^{-6}-13z^{2}-a^{3}z-4az-6za^{-1}-4za^{-3}-za^{-5}+1$

Vassiliev invariants

V2 and V3:

(0, 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 $0$ $0$ $0$ $0$ $0$ $0$ $32$ $32$ $0$ $0$ $0$ $0$ $0$ $32$ $176$ $-160$ $16$ $-48$

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=$2 is the signature of K11a81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   5 1   4 9                 8 3     -5 7               10 5       5 5             10 8         -2 3           10 10           0 1         8 11             3 -1       5 9               -4 -3     3 8                 5 -5   1 5                   -4 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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[11, Alternating, 81]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 81]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], X[22, 10, 1, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[20, 16, 21, 15], X[8, 17, 9, 18], X[6, 20, 7, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 81]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -5]
In[5]:=	BR[Knot[11, Alternating, 81]]
Out[5]=	BR[Knot[11, Alternating, 81]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 81]][t]
Out[6]=	-4 6 16 26 2 3 4 -29 - t + -- - -- + -- + 26 t - 16 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 81]][z]
Out[7]=	6 8 1 - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 81], Knot[11, Alternating, 282]}
In[9]:=	{KnotDet[Knot[11, Alternating, 81]], KnotSignature[Knot[11, Alternating, 81]]}
Out[9]=	{127, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 81]][q]
Out[10]=	-4 4 8 13 2 3 4 5 6 -17 - q + -- - -- + -- + 20 q - 20 q + 18 q - 13 q + 8 q - 4 q + 3 2 q q q 7 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 81], Knot[11, Alternating, 282]}
In[12]:=	A2Invariant[Knot[11, Alternating, 81]][q]
Out[12]=	-12 -10 -8 -6 3 3 2 4 6 8 1 - q + q + q - q + -- - -- + q - 2 q + 5 q - 3 q + 4 2 q q 10 12 14 16 18 20 3 q - q - 2 q + 2 q - 2 q + q
In[13]:=	Kauffman[Knot[11, Alternating, 81]][a, z]
Out[13]=	2 2 2 z 4 z 6 z 3 2 2 z z 9 z 1 - -- - --- - --- - 4 a z - a z - 13 z + ---- + -- - ---- - 5 3 a 6 4 2 a a a a a 3 3 3 3 2 2 2 z 5 z 19 z 18 z 3 3 3 4 5 a z - ---- + ---- + ----- + ----- + 9 a z + 3 a z + 44 z + 7 5 3 a a a a 4 4 4 4 5 5 5 z 7 z 2 z 39 z 2 4 4 z 13 z 23 z 5 -- - ---- + ---- + ----- + 15 a z + ---- - ----- - ----- + 3 a z - 8 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 3 5 6 8 z 14 z 50 z 2 6 11 z z 3 a z - 42 z + ---- - ----- - ----- - 14 a z + ----- - -- - 6 4 2 5 3 a a a a a 7 8 8 9 26 z 7 3 7 8 11 z 15 z 2 8 7 z ----- - 13 a z + a z + 8 z + ----- + ----- + 4 a z + ---- + a 4 2 3 a a a 9 10 12 z 9 10 2 z ----- + 5 a z + 2 z + ----- a 2 a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 81]], Vassiliev[3][Knot[11, Alternating, 81]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[11, Alternating, 81]][q, t]
Out[15]=	3 1 3 1 5 3 8 5 11 q + 10 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t 9 8 q 3 5 5 2 7 2 7 3 --- + --- + 10 q t + 10 q t + 8 q t + 10 q t + 5 q t + q t t 9 3 9 4 11 4 11 5 13 5 15 6 8 q t + 3 q t + 5 q t + q t + 3 q t + q t