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

K11a131 K11a133

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X16,6,17,5 X12,8,13,7 X18,9,19,10 X2,11,3,12 X20,14,21,13 X6,16,7,15 X22,18,1,17 X8,19,9,20 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 12 18 2 20 6 22 8 14

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, K11a352, ...}

Determinant and Signature:

{135, 2}

Jones Polynomial:

q-3 - 3q-2 + 8q-1 - 13 + 18q - 22q2 + 22q3 - 19q4 + 15q5 - 9q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 4q-4 - q-2 + 3q2 - 6q4 + q6 - 3q8 + q10 + 4q12 - 2q14 + 5q16 - 2q18 - 2q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6 - a-6z2 - a-6z4 + 4a-4 + 4a-4z2 + 2a-4z4 + a-4z6 - 4a-2 - 4a-2z2 + a-2z6 + 1 - 2z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 5a-8z4 + 4a-8z6 - a-7z + 5a-7z3 - 12a-7z5 + 8a-7z7 + a-6 - 5a-6z2 + 12a-6z4 - 17a-6z6 + 10a-6z8 - a-5z + 6a-5z3 - 5a-5z5 - 5a-5z7 + 7a-5z9 + 4a-4 - 21a-4z2 + 42a-4z4 - 40a-4z6 + 14a-4z8 + 2a-4z10 + 3a-3z - 8a-3z3 + 17a-3z5 - 23a-3z7 + 12a-3z9 + 4a-2 - 20a-2z2 + 33a-2z4 - 30a-2z6 + 9a-2z8 + 2a-2z10 + 3a-1z - 4a-1z3 + 2a-1z5 - 7a-1z7 + 5a-1z9 + 1 - 2z2 + 5z4 - 10z6 + 5z8 + 4az3 - 7az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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 11132. 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                 9 3     j = 9               10 6       j = 7             12 9         j = 5           10 10           j = 3         8 12             j = 1       6 11               j = -1     2 7                 j = -3   1 6                   j = -5   2                     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, 132]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 132]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 6, 17, 5], X[12, 8, 13, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[20, 14, 21, 13], X[6, 16, 7, 15], > X[22, 18, 1, 17], X[8, 19, 9, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 132]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -5, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 132]]
Out[5]=	DTCode[4, 10, 16, 12, 18, 2, 20, 6, 22, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 132]][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, 132]][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, 132]], KnotSignature[Knot[11, Alternating, 132]]}
Out[9]=	{135, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 132]][q]
Out[10]=	-3 3 8 2 3 4 5 6 7 8 -13 + q - -- + - + 18 q - 22 q + 22 q - 19 q + 15 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, 132]}
In[12]:=	A2Invariant[Knot[11, Alternating, 132]][q]
Out[12]=	-10 -6 4 -2 2 4 6 8 10 12 14 16 q - q + -- - q + 3 q - 6 q + q - 3 q + q + 4 q - 2 q + 5 q - 4 q 18 20 22 24 > 2 q - 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 132]][a, z]
Out[13]=	2 2 2 4 4 -6 4 4 2 2 z 4 z 4 z 2 2 4 z 2 z 1 - a + -- - -- + a - 2 z - -- + ---- - ---- + a z - 2 z - -- + ---- + 4 2 6 4 2 6 4 a a a a a a a 6 6 z z > -- + -- 4 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 132]][a, z]
Out[14]=	2 2 2 -6 4 4 2 z z 3 z 3 z 2 z 5 z 21 z 1 + a + -- + -- - a - -- - -- + --- + --- - 2 z + -- - ---- - ----- - 4 2 7 5 3 a 8 6 4 a a a a a a a a 2 3 3 3 3 3 4 20 z 2 2 z 5 z 6 z 8 z 4 z 3 4 5 z > ----- + 3 a z - -- + ---- + ---- - ---- - ---- + 4 a z + 5 z - ---- + 2 9 7 5 3 a 8 a a a a a a 4 4 4 5 5 5 5 5 12 z 42 z 33 z 2 4 z 12 z 5 z 17 z 2 z > ----- + ----- + ----- - 3 a z + -- - ----- - ---- + ----- + ---- - 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 5 6 4 z 17 z 40 z 30 z 2 6 8 z 5 z > 7 a z - 10 z + ---- - ----- - ----- - ----- + a z + ---- - ---- - 8 6 4 2 7 5 a a a a a a 7 7 8 8 8 9 9 9 23 z 7 z 7 8 10 z 14 z 9 z 7 z 12 z 5 z > ----- - ---- + 3 a z + 5 z + ----- + ----- + ---- + ---- + ----- + ---- + 3 a 6 4 2 5 3 a a a a a a a 10 10 2 z 2 z > ----- + ----- 4 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 132]], Vassiliev[3][Knot[11, Alternating, 132]]}
Out[15]=	{-2, 0}
In[16]:=	Kh[Knot[11, Alternating, 132]][q, t]
Out[16]=	3 1 2 1 6 2 7 6 q 3 11 q + 8 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 12 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 + 10 q t + 12 q t + 9 q t + 10 q t + 6 q t + 9 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 K11a132

K11a131 K11a133