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

K11a187 K11a189

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 20 18 2 6 22 10 8

Alexander Polynomial:

2t-3 - 9t-2 + 15t-1 - 15 + 15t - 9t2 + 2t3

Conway Polynomial:

1 - 3z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{67, 2}

Jones Polynomial:

q-5 - 2q-4 + 4q-3 - 6q-2 + 8q-1 - 10 + 10q - 9q2 + 8q3 - 5q4 + 3q5 - q6

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

{...}

A2 (sl(3)) Invariant:

q-16 + q-14 + q-10 - q-8 - q-6 - 2q-2 + 1 + 2q6 - q8 + 2q10 + q16 - q18

HOMFLY-PT Polynomial:

- 2a-4z2 - a-4z4 + a-2 + 2a-2z2 + 3a-2z4 + a-2z6 + 1 + 2z2 + 3z4 + z6 - 3a2 - 6a2z2 - 2a2z4 + 2a4 + a4z2

Kauffman Polynomial:

a-7z3 - a-6z2 + 3a-6z4 - 3a-5z3 + 5a-5z5 + 4a-4z2 - 10a-4z4 + 7a-4z6 - 2a-3z + 6a-3z3 - 13a-3z5 + 7a-3z7 - a-2 + 8a-2z2 - 8a-2z4 - 7a-2z6 + 5a-2z8 - 5a-1z + 15a-1z3 - 13a-1z5 - 3a-1z7 + 3a-1z9 + 1 - 7z2 + 23z4 - 21z6 + 3z8 + z10 - az - 6az3 + 24az5 - 21az7 + 5az9 + 3a2 - 19a2z2 + 30a2z4 - 13a2z6 - a2z8 + a2z10 + 2a3z - 11a3z3 + 19a3z5 - 11a3z7 + 2a3z9 + 2a4 - 9a4z2 + 12a4z4 - 6a4z6 + a4z8

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

{-3, 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=2 is the signature of 11188. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                       1 j = 11                     2   j = 9                   3 1   j = 7                 5 2     j = 5               4 3       j = 3             6 5         j = 1           5 5           j = -1         3 5             j = -3       3 5               j = -5     1 3                 j = -7   1 3                   j = -9   1                     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, 188]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 188]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[14, 6, 15, 5], X[16, 8, 17, 7], > X[20, 9, 21, 10], X[18, 11, 19, 12], X[2, 13, 3, 14], X[6, 16, 7, 15], > X[22, 18, 1, 17], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 188]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -6, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 188]]
Out[5]=	DTCode[4, 12, 14, 16, 20, 18, 2, 6, 22, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 188]][t]
Out[6]=	2 9 15 2 3 -15 + -- - -- + -- + 15 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 188]][z]
Out[7]=	2 4 6 1 - 3 z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 188]}
In[9]:=	{KnotDet[Knot[11, Alternating, 188]], KnotSignature[Knot[11, Alternating, 188]]}
Out[9]=	{67, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 188]][q]
Out[10]=	-5 2 4 6 8 2 3 4 5 6 -10 + q - -- + -- - -- + - + 10 q - 9 q + 8 q - 5 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, 188]}
In[12]:=	A2Invariant[Knot[11, Alternating, 188]][q]
Out[12]=	-16 -14 -10 -8 -6 2 6 8 10 16 18 1 + q + q + q - q - q - -- + 2 q - q + 2 q + q - q 2 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 188]][a, z]
Out[13]=	2 2 4 -2 2 4 2 2 z 2 z 2 2 4 2 4 z 1 + a - 3 a + 2 a + 2 z - ---- + ---- - 6 a z + a z + 3 z - -- + 4 2 4 a a a 4 6 3 z 2 4 6 z > ---- - 2 a z + z + -- 2 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 188]][a, z]
Out[14]=	2 2 2 -2 2 4 2 z 5 z 3 2 z 4 z 8 z 1 - a + 3 a + 2 a - --- - --- - a z + 2 a z - 7 z - -- + ---- + ---- - 3 a 6 4 2 a a a a 3 3 3 3 2 2 4 2 z 3 z 6 z 15 z 3 3 3 4 > 19 a z - 9 a z + -- - ---- + ---- + ----- - 6 a z - 11 a z + 23 z + 7 5 3 a a a a 4 4 4 5 5 5 3 z 10 z 8 z 2 4 4 4 5 z 13 z 13 z > ---- - ----- - ---- + 30 a z + 12 a z + ---- - ----- - ----- + 6 4 2 5 3 a a a a a a 6 6 7 5 3 5 6 7 z 7 z 2 6 4 6 7 z > 24 a z + 19 a z - 21 z + ---- - ---- - 13 a z - 6 a z + ---- - 4 2 3 a a a 7 8 9 3 z 7 3 7 8 5 z 2 8 4 8 3 z 9 > ---- - 21 a z - 11 a z + 3 z + ---- - a z + a z + ---- + 5 a z + a 2 a a 3 9 10 2 10 > 2 a z + z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 188]], Vassiliev[3][Knot[11, Alternating, 188]]}
Out[15]=	{-3, 2}
In[16]:=	Kh[Knot[11, Alternating, 188]][q, t]
Out[16]=	3 1 1 1 3 1 3 3 5 5 q + 6 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 11 6 9 5 7 5 7 4 5 4 5 3 3 3 3 2 q t q t q t q t q t q t q t q t 3 5 5 q 3 5 5 2 7 2 7 3 > ---- + --- + --- + 5 q t + 4 q t + 3 q t + 5 q t + 2 q t + 2 q t t q t 9 3 9 4 11 4 13 5 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a188

K11a187 K11a189