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

K11a354 K11a356

Knotscape

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DrawMorseLink

PD Presentation:

X8291 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X12,20,13,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

8 14 16 18 20 22 2 4 6 12 10

Alexander Polynomial:

2t-4 - 4t-3 + 6t-2 - 7t-1 + 7 - 7t + 6t2 - 4t3 + 2t4

Conway Polynomial:

1 + 13z2 + 22z4 + 12z6 + 2z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{45, 8}

Jones Polynomial:

q4 - q5 + 3q6 - 4q7 + 5q8 - 6q9 + 7q10 - 6q11 + 5q12 - 4q13 + 2q14 - q15

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

{...}

A2 (sl(3)) Invariant:

q14 + 2q18 + q22 + 2q28 + 2q32 - q34 - q36 - q38 - q40 - q44

HOMFLY-PT Polynomial:

- 3a-12 - 7a-12z2 - 5a-12z4 - a-12z6 + 2a-10 + 7a-10z2 + 11a-10z4 + 6a-10z6 + a-10z8 + 2a-8 + 13a-8z2 + 16a-8z4 + 7a-8z6 + a-8z8

Kauffman Polynomial:

- a-19z + a-19z3 - a-18z2 + 2a-18z4 + 2a-17z - 3a-17z3 + 3a-17z5 + a-16z2 - 3a-16z4 + 3a-16z6 - a-15z + 2a-15z3 - 5a-15z5 + 3a-15z7 - 7a-14z2 + 10a-14z4 - 9a-14z6 + 3a-14z8 - 2a-13z3 + 4a-13z5 - 6a-13z7 + 2a-13z9 - 3a-12 + 11a-12z2 - 9a-12z4 + 3a-12z6 - 3a-12z8 + a-12z10 + 3a-11z - 8a-11z3 + 18a-11z5 - 14a-11z7 + 3a-11z9 - 2a-10 + 7a-10z2 - 8a-10z4 + 8a-10z6 - 5a-10z8 + a-10z10 - a-9z + 6a-9z5 - 5a-9z7 + a-9z9 + 2a-8 - 13a-8z2 + 16a-8z4 - 7a-8z6 + a-8z8

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

{13, 45}

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 11355. 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                     1   j = 27                   3 1   j = 25                 2 1     j = 23               4 3       j = 21             3 2         j = 19           3 4           j = 17         2 3             j = 15       2 3               j = 13     1 2                 j = 11     2                   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, 355]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 355]]
Out[3]=	PD[X[8, 2, 9, 1], X[14, 4, 15, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[20, 10, 21, 9], X[22, 12, 1, 11], X[2, 14, 3, 13], X[4, 16, 5, 15], > X[6, 18, 7, 17], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 355]]
Out[4]=	GaussCode[1, -7, 2, -8, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 355]]
Out[5]=	DTCode[8, 14, 16, 18, 20, 22, 2, 4, 6, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 355]][t]
Out[6]=	2 4 6 7 2 3 4 7 + -- - -- + -- - - - 7 t + 6 t - 4 t + 2 t 4 3 2 t t t t
In[7]:=	Conway[Knot[11, Alternating, 355]][z]
Out[7]=	2 4 6 8 1 + 13 z + 22 z + 12 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 355]}
In[9]:=	{KnotDet[Knot[11, Alternating, 355]], KnotSignature[Knot[11, Alternating, 355]]}
Out[9]=	{45, 8}
In[10]:=	J=Jones[Knot[11, Alternating, 355]][q]
Out[10]=	4 5 6 7 8 9 10 11 12 13 14 q - q + 3 q - 4 q + 5 q - 6 q + 7 q - 6 q + 5 q - 4 q + 2 q - 15 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 355]}
In[12]:=	A2Invariant[Knot[11, Alternating, 355]][q]
Out[12]=	14 18 22 28 32 34 36 38 40 44 q + 2 q + q + 2 q + 2 q - q - q - q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 355]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 -3 2 2 7 z 7 z 13 z 5 z 11 z 16 z z 6 z --- + --- + -- - ---- + ---- + ----- - ---- + ----- + ----- - --- + ---- + 12 10 8 12 10 8 12 10 8 12 10 a 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, 355]][a, z]
Out[14]=	2 2 2 2 2 -3 2 2 z 2 z z 3 z z z z 7 z 11 z 7 z --- - --- + -- - --- + --- - --- + --- - -- - --- + --- - ---- + ----- + ---- - 12 10 8 19 17 15 11 9 18 16 14 12 10 a a a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 13 z z 3 z 2 z 2 z 8 z 2 z 3 z 10 z 9 z > ----- + --- - ---- + ---- - ---- - ---- + ---- - ---- + ----- - ---- - 8 19 17 15 13 11 18 16 14 12 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 8 z 16 z 3 z 5 z 4 z 18 z 6 z 3 z 9 z 3 z > ---- + ----- + ---- - ---- + ---- + ----- + ---- + ---- - ---- + ---- + 10 8 17 15 13 11 9 16 14 12 a a a a a a a a a a 6 6 7 7 7 7 8 8 8 8 9 8 z 7 z 3 z 6 z 14 z 5 z 3 z 3 z 5 z z 2 z > ---- - ---- + ---- - ---- - ----- - ---- + ---- - ---- - ---- + -- + ---- + 10 8 15 13 11 9 14 12 10 8 13 a a a a a a a a a a a 9 9 10 10 3 z z z z > ---- + -- + --- + --- 11 9 12 10 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 355]], Vassiliev[3][Knot[11, Alternating, 355]]}
Out[15]=	{13, 45}
In[16]:=	Kh[Knot[11, Alternating, 355]][q, t]
Out[16]=	7 9 9 11 2 13 2 13 3 15 3 15 4 q + q + q t + 2 q t + q t + 2 q t + 2 q t + 3 q t + 17 4 17 5 19 5 19 6 21 6 21 7 > 2 q t + 3 q t + 3 q t + 4 q t + 3 q t + 2 q t + 23 7 23 8 25 8 25 9 27 9 27 10 29 10 > 4 q t + 3 q t + 2 q t + q t + 3 q t + q t + q t + 31 11 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a355

K11a354 K11a356