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

K11a351 K11a353

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X18,4,19,3 X16,5,17,6 X14,8,15,7 X20,9,21,10 X4,12,5,11 X2,13,3,14 X22,16,1,15 X12,18,13,17 X10,19,11,20 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 18 16 14 20 4 2 22 12 10 8

Alexander Polynomial:

2t-3 - 13t-2 + 32t-1 - 41 + 32t - 13t2 + 2t3

Conway Polynomial:

1 - 2z2 - z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a6, K11a132, ...}

Determinant and Signature:

{135, 2}

Jones Polynomial:

- q-4 + 5q-3 - 9q-2 + 14q-1 - 19 + 21q - 21q2 + 19q3 - 13q4 + 8q5 - 4q6 + q7

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

{...}

A2 (sl(3)) Invariant:

- q-12 + 3q-10 + 4q-4 - 5q-2 - q2 - 2q4 + 4q6 - 2q8 + 4q10 - 2q14 + 3q16 - 2q18 - q20 + q22

HOMFLY-PT Polynomial:

a-6z2 - 2a-4z2 - 2a-4z4 + 2a-2 + a-2z2 + a-2z4 + a-2z6 - 3 - 2z2 + z4 + z6 + 2a2 - a2z4

Kauffman Polynomial:

a-8z4 - 2a-7z3 + 4a-7z5 + 2a-6z2 - 7a-6z4 + 8a-6z6 + 2a-5z3 - 12a-5z5 + 11a-5z7 + 2a-4z2 + 3a-4z4 - 17a-4z6 + 12a-4z8 - 2a-3z3 + 6a-3z5 - 17a-3z7 + 10a-3z9 - 2a-2 + 28a-2z4 - 37a-2z6 + 6a-2z8 + 4a-2z10 - 2a-1z - 12a-1z3 + 51a-1z5 - 57a-1z7 + 18a-1z9 - 3 + 2z2 + 28z4 - 28z6 - z8 + 4z10 - 2az - 6az3 + 27az5 - 28az7 + 8az9 - 2a2 + 2a2z2 + 11a2z4 - 16a2z6 + 5a2z8 - 2a3z5 + a3z7

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

{-2, 0}

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 11352. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   5 1   j = 9                 8 3     j = 7               11 5       j = 5             10 8         j = 3           11 11           j = 1         9 11             j = -1       5 10               j = -3     4 9                 j = -5   1 5                   j = -7   4                     j = -9 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, 352]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 352]]
Out[3]=	PD[X[6, 2, 7, 1], X[18, 4, 19, 3], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[20, 9, 21, 10], X[4, 12, 5, 11], X[2, 13, 3, 14], X[22, 16, 1, 15], > X[12, 18, 13, 17], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 352]]
Out[4]=	GaussCode[1, -7, 2, -6, 3, -1, 4, -11, 5, -10, 6, -9, 7, -4, 8, -3, 9, -2, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 352]]
Out[5]=	DTCode[6, 18, 16, 14, 20, 4, 2, 22, 12, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 352]][t]
Out[6]=	2 13 32 2 3 -41 + -- - -- + -- + 32 t - 13 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 352]][z]
Out[7]=	2 4 6 1 - 2 z - z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 6], Knot[11, Alternating, 132], > Knot[11, Alternating, 352]}
In[9]:=	{KnotDet[Knot[11, Alternating, 352]], KnotSignature[Knot[11, Alternating, 352]]}
Out[9]=	{135, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 352]][q]
Out[10]=	-4 5 9 14 2 3 4 5 6 7 -19 - q + -- - -- + -- + 21 q - 21 q + 19 q - 13 q + 8 q - 4 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 352]}
In[12]:=	A2Invariant[Knot[11, Alternating, 352]][q]
Out[12]=	-12 3 4 5 2 4 6 8 10 14 16 -q + --- + -- - -- - q - 2 q + 4 q - 2 q + 4 q - 2 q + 3 q - 10 4 2 q q q 18 20 22 > 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 352]][a, z]
Out[13]=	2 2 2 4 4 6 2 2 2 z 2 z z 4 2 z z 2 4 6 z -3 + -- + 2 a - 2 z + -- - ---- + -- + z - ---- + -- - a z + z + -- 2 6 4 2 4 2 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 352]][a, z]
Out[14]=	2 2 3 3 2 2 2 z 2 2 z 2 z 2 2 2 z 2 z -3 - -- - 2 a - --- - 2 a z + 2 z + ---- + ---- + 2 a z - ---- + ---- - 2 a 6 4 7 5 a a a a a 3 3 4 4 4 4 2 z 12 z 3 4 z 7 z 3 z 28 z 2 4 > ---- - ----- - 6 a z + 28 z + -- - ---- + ---- + ----- + 11 a z + 3 a 8 6 4 2 a a a a a 5 5 5 5 6 6 4 z 12 z 6 z 51 z 5 3 5 6 8 z 17 z > ---- - ----- + ---- + ----- + 27 a z - 2 a z - 28 z + ---- - ----- - 7 5 3 a 6 4 a a a a a 6 7 7 7 8 37 z 2 6 11 z 17 z 57 z 7 3 7 8 12 z > ----- - 16 a z + ----- - ----- - ----- - 28 a z + a z - z + ----- + 2 5 3 a 4 a a a a 8 9 9 10 6 z 2 8 10 z 18 z 9 10 4 z > ---- + 5 a z + ----- + ----- + 8 a z + 4 z + ----- 2 3 a 2 a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 352]], Vassiliev[3][Knot[11, Alternating, 352]]}
Out[15]=	{-2, 0}
In[16]:=	Kh[Knot[11, Alternating, 352]][q, t]
Out[16]=	3 1 4 1 5 4 9 5 10 11 q + 11 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 q t 9 q 3 5 5 2 7 2 7 3 9 3 > --- + 11 q t + 10 q t + 8 q t + 11 q t + 5 q t + 8 q t + t 9 4 11 4 11 5 13 5 15 6 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a352

K11a351 K11a353