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

K11a262 K11a264

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8493 X16,6,17,5 X2837 X20,10,21,9 X22,12,1,11 X18,14,19,13 X4,16,5,15 X12,18,13,17 X14,20,15,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 16 2 20 22 18 4 12 14 10

Alexander Polynomial:

2t-4 - 6t-3 + 11t-2 - 14t-1 + 15 - 14t + 11t2 - 6t3 + 2t4

Conway Polynomial:

1 + 8z2 + 15z4 + 10z6 + 2z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{81, 8}

Jones Polynomial:

q4 - q5 + 5q6 - 7q7 + 10q8 - 13q9 + 12q10 - 12q11 + 10q12 - 6q13 + 3q14 - q15

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

{...}

A2 (sl(3)) Invariant:

q14 + 4q18 + q20 + 4q22 + q24 - 2q26 - q28 - 7q30 - 2q34 + 2q36 + 3q38 + q42 - 2q44

HOMFLY-PT Polynomial:

- a-14 + 5a-12 + a-12z2 - 3a-12z4 - a-12z6 - 11a-10 - 13a-10z2 + 4a-10z6 + a-10z8 + 8a-8 + 20a-8z2 + 18a-8z4 + 7a-8z6 + a-8z8

Kauffman Polynomial:

a-19z3 + 3a-18z4 - 3a-17z3 + 6a-17z5 + 6a-16z2 - 15a-16z4 + 10a-16z6 - 6a-15z + 18a-15z3 - 27a-15z5 + 12a-15z7 + a-14 + 5a-14z2 - 18a-14z6 + 9a-14z8 - 16a-13z + 43a-13z3 - 36a-13z5 + a-13z7 + 4a-13z9 + 5a-12 - 7a-12z2 + 28a-12z4 - 32a-12z6 + 7a-12z8 + a-12z10 - 22a-11z + 37a-11z3 - 6a-11z5 - 14a-11z7 + 5a-11z9 + 11a-10 - 26a-10z2 + 28a-10z4 - 11a-10z6 - a-10z8 + a-10z10 - 12a-9z + 16a-9z3 - 3a-9z5 - 3a-9z7 + a-9z9 + 8a-8 - 20a-8z2 + 18a-8z4 - 7a-8z6 + a-8z8

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

{8, 21}

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=8 is the signature of 11263. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 31                       1 j = 29                     2   j = 27                   4 1   j = 25                 6 2     j = 23               6 4       j = 21             6 6         j = 19           7 6           j = 17         3 6             j = 15       4 7               j = 13     1 3                 j = 11     4                   j = 9 1 1                     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, 263]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 263]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[16, 6, 17, 5], X[2, 8, 3, 7], > X[20, 10, 21, 9], X[22, 12, 1, 11], X[18, 14, 19, 13], X[4, 16, 5, 15], > X[12, 18, 13, 17], X[14, 20, 15, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 263]]
Out[4]=	GaussCode[1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -9, 7, -10, 8, -3, 9, -7, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 263]]
Out[5]=	DTCode[6, 8, 16, 2, 20, 22, 18, 4, 12, 14, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 263]][t]
Out[6]=	2 6 11 14 2 3 4 15 + -- - -- + -- - -- - 14 t + 11 t - 6 t + 2 t 4 3 2 t t t t
In[7]:=	Conway[Knot[11, Alternating, 263]][z]
Out[7]=	2 4 6 8 1 + 8 z + 15 z + 10 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 263]}
In[9]:=	{KnotDet[Knot[11, Alternating, 263]], KnotSignature[Knot[11, Alternating, 263]]}
Out[9]=	{81, 8}
In[10]:=	J=Jones[Knot[11, Alternating, 263]][q]
Out[10]=	4 5 6 7 8 9 10 11 12 13 q - q + 5 q - 7 q + 10 q - 13 q + 12 q - 12 q + 10 q - 6 q + 14 15 > 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 263]}
In[12]:=	A2Invariant[Knot[11, Alternating, 263]][q]
Out[12]=	14 18 20 22 24 26 28 30 34 36 38 q + 4 q + q + 4 q + q - 2 q - q - 7 q - 2 q + 2 q + 3 q + 42 44 > q - 2 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 263]][a, z]
Out[13]=	2 2 2 4 4 6 6 -14 5 11 8 z 13 z 20 z 3 z 18 z z 4 z -a + --- - --- + -- + --- - ----- + ----- - ---- + ----- - --- + ---- + 12 10 8 12 10 8 12 8 12 10 a a a a a a a a a a 6 8 8 7 z z z > ---- + --- + -- 8 10 8 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 263]][a, z]
Out[14]=	2 2 2 2 -14 5 11 8 6 z 16 z 22 z 12 z 6 z 5 z 7 z 26 z a + --- + --- + -- - --- - ---- - ---- - ---- + ---- + ---- - ---- - ----- - 12 10 8 15 13 11 9 16 14 12 10 a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 20 z z 3 z 18 z 43 z 37 z 16 z 3 z 15 z 28 z > ----- + --- - ---- + ----- + ----- + ----- + ----- + ---- - ----- + ----- + 8 19 17 15 13 11 9 18 16 12 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 28 z 18 z 6 z 27 z 36 z 6 z 3 z 10 z 18 z > ----- + ----- + ---- - ----- - ----- - ---- - ---- + ----- - ----- - 10 8 17 15 13 11 9 16 14 a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 32 z 11 z 7 z 12 z z 14 z 3 z 9 z 7 z z > ----- - ----- - ---- + ----- + --- - ----- - ---- + ---- + ---- - --- + 12 10 8 15 13 11 9 14 12 10 a a a a a a a a a a 8 9 9 9 10 10 z 4 z 5 z z z z > -- + ---- + ---- + -- + --- + --- 8 13 11 9 12 10 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 263]], Vassiliev[3][Knot[11, Alternating, 263]]}
Out[15]=	{8, 21}
In[16]:=	Kh[Knot[11, Alternating, 263]][q, t]
Out[16]=	7 9 9 11 2 13 2 13 3 15 3 15 4 q + q + q t + 4 q t + q t + 3 q t + 4 q t + 7 q t + 17 4 17 5 19 5 19 6 21 6 21 7 > 3 q t + 6 q t + 7 q t + 6 q t + 6 q t + 6 q t + 23 7 23 8 25 8 25 9 27 9 27 10 > 6 q t + 4 q t + 6 q t + 2 q t + 4 q t + q t + 29 10 31 11 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a263

K11a262 K11a264