© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a263 K11a265

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X16,6,17,5 X14,7,15,8 X4,9,5,10 X18,12,19,11 X20,14,21,13 X2,16,3,15 X22,17,1,18 X12,20,13,19 X10,22,11,21

Gauss Code:

{1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -4, 8, -3, 9, -6, 10, -7, 11, -9}

DT (Dowker-Thistlethwaite) Code:

6 8 16 14 4 18 20 2 22 12 10

Alexander Polynomial:

- t-4 + 6t-3 - 16t-2 + 28t-1 - 33 + 28t - 16t2 + 6t3 - t4

Conway Polynomial:

1 + 2z2 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a157, K11a305, ...}

Determinant and Signature:

{135, 2}

Jones Polynomial:

q-3 - 4q-2 + 8q-1 - 13 + 19q - 21q2 + 22q3 - 19q4 + 14q5 - 9q6 + 4q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-8 - 2q-6 + 2q-4 - 2q-2 - 1 + 4q2 - 3q4 + 6q6 - q8 + q10 + q12 - 4q14 + 3q16 - 2q18 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + a-4 + 7a-4z2 + 7a-4z4 + 2a-4z6 + a-2 - 5a-2z2 - 9a-2z4 - 5a-2z6 - a-2z8 + 2z2 + 3z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 5a-8z4 + 4a-8z6 - 2a-7z + 7a-7z3 - 13a-7z5 + 8a-7z7 + a-6 - 4a-6z2 + 7a-6z4 - 13a-6z6 + 9a-6z8 - 2a-5z + 11a-5z3 - 14a-5z5 + 6a-5z9 + a-4 - 13a-4z2 + 34a-4z4 - 36a-4z6 + 13a-4z8 + 2a-4z10 + 2a-3z3 + 10a-3z5 - 22a-3z7 + 12a-3z9 - a-2 - 13a-2z2 + 40a-2z4 - 39a-2z6 + 11a-2z8 + 2a-2z10 + 4a-1z3 - 10a-1z7 + 6a-1z9 - 5z2 + 16z4 - 19z6 + 7z8 + 5az3 - 10az5 + 4az7 - 2a2z4 + a2z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{2, 3}

Khovanov Homology: (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 11264. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   6 1   j = 11                 8 3     j = 9               11 6       j = 7             11 8         j = 5           10 11           j = 3         9 11             j = 1       5 11               j = -1     3 8                 j = -3   1 5                   j = -5   3                     j = -7 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 264]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 264]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[16, 6, 17, 5], X[14, 7, 15, 8], > X[4, 9, 5, 10], X[18, 12, 19, 11], X[20, 14, 21, 13], X[2, 16, 3, 15], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 264]]
Out[4]=	GaussCode[1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -4, 8, -3, 9, -6, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 264]]
Out[5]=	DTCode[6, 8, 16, 14, 4, 18, 20, 2, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 264]][t]
Out[6]=	-4 6 16 28 2 3 4 -33 - t + -- - -- + -- + 28 t - 16 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 264]][z]
Out[7]=	2 6 8 1 + 2 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 157], Knot[11, Alternating, 264], > Knot[11, Alternating, 305]}
In[9]:=	{KnotDet[Knot[11, Alternating, 264]], KnotSignature[Knot[11, Alternating, 264]]}
Out[9]=	{135, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 264]][q]
Out[10]=	-3 4 8 2 3 4 5 6 7 8 -13 + q - -- + - + 19 q - 21 q + 22 q - 19 q + 14 q - 9 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 264]}
In[12]:=	A2Invariant[Knot[11, Alternating, 264]][q]
Out[12]=	-8 2 2 2 2 4 6 8 10 12 14 16 -1 + q - -- + -- - -- + 4 q - 3 q + 6 q - q + q + q - 4 q + 3 q - 6 4 2 q q q 18 22 24 > 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 264]][a, z]
Out[13]=	2 2 2 4 4 4 -6 -4 -2 2 2 z 7 z 5 z 4 z 7 z 9 z 6 -a + a + a + 2 z - ---- + ---- - ---- + 3 z - -- + ---- - ---- + z + 6 4 2 6 4 2 a a a a a a 6 6 8 2 z 5 z z > ---- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 264]][a, z]
Out[14]=	2 2 2 2 3 3 -6 -4 -2 2 z 2 z 2 z 4 z 13 z 13 z z 7 z a + a - a - --- - --- - 5 z + -- - ---- - ----- - ----- - -- + ---- + 7 5 8 6 4 2 9 7 a a a a a a a a 3 3 3 4 4 4 4 11 z 2 z 4 z 3 4 5 z 7 z 34 z 40 z > ----- + ---- + ---- + 5 a z + 16 z - ---- + ---- + ----- + ----- - 5 3 a 8 6 4 2 a a a a a a 5 5 5 5 6 6 2 4 z 13 z 14 z 10 z 5 6 4 z 13 z > 2 a z + -- - ----- - ----- + ----- - 10 a z - 19 z + ---- - ----- - 9 7 5 3 8 6 a a a a a a 6 6 7 7 7 8 36 z 39 z 2 6 8 z 22 z 10 z 7 8 9 z > ----- - ----- + a z + ---- - ----- - ----- + 4 a z + 7 z + ---- + 4 2 7 3 a 6 a a a a a 8 8 9 9 9 10 10 13 z 11 z 6 z 12 z 6 z 2 z 2 z > ----- + ----- + ---- + ----- + ---- + ----- + ----- 4 2 5 3 a 4 2 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 264]], Vassiliev[3][Knot[11, Alternating, 264]]}
Out[15]=	{2, 3}
In[16]:=	Kh[Knot[11, Alternating, 264]][q, t]
Out[16]=	3 1 3 1 5 3 8 5 q 3 11 q + 9 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 11 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 10 q t + 11 q t + 11 q t + 8 q t + 11 q t + 6 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a264

K11a263 K11a265