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

K11a86 K11a88

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 18 2 20 8 6 22 14

Alexander Polynomial:

- 2t-3 + 11t-2 - 28t-1 + 39 - 28t + 11t2 - 2t3

Conway Polynomial:

1 - 2z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{121, 0}

Jones Polynomial:

- q-5 + 4q-4 - 8q-3 + 13q-2 - 17q-1 + 20 - 19q + 16q2 - 12q3 + 7q4 - 3q5 + q6

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

{K11a28, K11a96, ...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 3q-10 + 2q-8 - 2q-4 + 5q-2 - 1 + 3q2 - 2q4 - 3q6 + 2q8 - 4q10 + 2q12 + 2q14 - q16 + q18

HOMFLY-PT Polynomial:

2a-4 + 2a-4z2 + a-4z4 - 4a-2 - 6a-2z2 - 3a-2z4 - a-2z6 + 3 + z2 - z4 - z6 + 2a2z2 + 2a2z4 - a4z2

Kauffman Polynomial:

2a-6z2 - 3a-6z4 + a-6z6 - 2a-5z + 6a-5z3 - 8a-5z5 + 3a-5z7 + 2a-4 - 7a-4z2 + 12a-4z4 - 13a-4z6 + 5a-4z8 - 3a-3z + 8a-3z3 - 6a-3z5 - 4a-3z7 + 4a-3z9 + 4a-2 - 21a-2z2 + 41a-2z4 - 37a-2z6 + 12a-2z8 + a-2z10 - 3a-1z + 12a-1z3 - 8a-1z5 - 8a-1z7 + 8a-1z9 + 3 - 13z2 + 31z4 - 34z6 + 14z8 + z10 - 3az + 16az3 - 22az5 + 6az7 + 4az9 + a2z2 - a2z4 - 7a2z6 + 7a2z8 - a3z + 5a3z3 - 11a3z5 + 7a3z7 + 2a4z2 - 6a4z4 + 4a4z6 - a5z3 + a5z5

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

{-2, -2}

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=0 is the signature of 1187. 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 = 13                       1 j = 11                     2   j = 9                   5 1   j = 7                 7 2     j = 5               9 5       j = 3             10 7         j = 1           10 9           j = -1         8 11             j = -3       5 9               j = -5     3 8                 j = -7   1 5                   j = -9   3                     j = -11 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, 87]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 87]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[2, 11, 3, 12], X[20, 13, 21, 14], X[8, 16, 9, 15], > X[6, 18, 7, 17], X[22, 19, 1, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 87]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, > -7, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 87]]
Out[5]=	DTCode[4, 10, 12, 16, 18, 2, 20, 8, 6, 22, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 87]][t]
Out[6]=	2 11 28 2 3 39 - -- + -- - -- - 28 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 87]][z]
Out[7]=	2 4 6 1 - 2 z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 87]}
In[9]:=	{KnotDet[Knot[11, Alternating, 87]], KnotSignature[Knot[11, Alternating, 87]]}
Out[9]=	{121, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 87]][q]
Out[10]=	-5 4 8 13 17 2 3 4 5 6 20 - q + -- - -- + -- - -- - 19 q + 16 q - 12 q + 7 q - 3 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 28], Knot[11, Alternating, 87], > Knot[11, Alternating, 96]}
In[12]:=	A2Invariant[Knot[11, Alternating, 87]][q]
Out[12]=	-16 -14 2 3 2 2 5 2 4 6 8 -1 - q + q + --- - --- + -- - -- + -- + 3 q - 2 q - 3 q + 2 q - 12 10 8 4 2 q q q q q 10 12 14 16 18 > 4 q + 2 q + 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 87]][a, z]
Out[13]=	2 2 4 4 2 4 2 2 z 6 z 2 2 4 2 4 z 3 z 2 4 3 + -- - -- + z + ---- - ---- + 2 a z - a z - z + -- - ---- + 2 a z - 4 2 4 2 4 2 a a a a a a 6 6 z > z - -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 87]][a, z]
Out[14]=	2 2 2 2 4 2 z 3 z 3 z 3 2 2 z 7 z 21 z 3 + -- + -- - --- - --- - --- - 3 a z - a z - 13 z + ---- - ---- - ----- + 4 2 5 3 a 6 4 2 a a a a a a a 3 3 3 2 2 4 2 6 z 8 z 12 z 3 3 3 5 3 4 > a z + 2 a z + ---- + ---- + ----- + 16 a z + 5 a z - a z + 31 z - 5 3 a a a 4 4 4 5 5 5 3 z 12 z 41 z 2 4 4 4 8 z 6 z 8 z 5 > ---- + ----- + ----- - a z - 6 a z - ---- - ---- - ---- - 22 a z - 6 4 2 5 3 a a a a a a 6 6 6 7 3 5 5 5 6 z 13 z 37 z 2 6 4 6 3 z > 11 a z + a z - 34 z + -- - ----- - ----- - 7 a z + 4 a z + ---- - 6 4 2 5 a a a a 7 7 8 8 9 4 z 8 z 7 3 7 8 5 z 12 z 2 8 4 z > ---- - ---- + 6 a z + 7 a z + 14 z + ---- + ----- + 7 a z + ---- + 3 a 4 2 3 a a a a 9 10 8 z 9 10 z > ---- + 4 a z + z + --- a 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 87]], Vassiliev[3][Knot[11, Alternating, 87]]}
Out[15]=	{-2, -2}
In[16]:=	Kh[Knot[11, Alternating, 87]][q, t]
Out[16]=	11 1 3 1 5 3 8 5 9 -- + 10 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 8 3 3 2 5 2 5 3 7 3 7 4 > --- + 9 q t + 10 q t + 7 q t + 9 q t + 5 q t + 7 q t + 2 q t + q t 9 4 9 5 11 5 13 6 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a87

K11a86 K11a88