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L11a213 L11a215

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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X22,20,7,19 X16,5,17,6 X18,10,19,9 X4,15,5,16 X10,18,11,17 X14,22,15,21 X20,14,21,13 X2738 X6,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 8q-9/2 + 13q-7/2 - 19q-5/2 + 21q-3/2 - 22q-1/2 + 19q1/2 - 14q3/2 + 9q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 + 4q-14 - 2q-12 + 4q-10 + 4q-8 - q-6 + 5q-4 - 4q-2 + 2 - 2q2 - 3q4 + 3q6 - 3q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 + a-1z-1 - a-1z - 4a-1z3 - 2a-1z5 - 2az-1 - az + 3az3 + 3az5 + az7 - 4a3z - 5a3z3 - 2a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + a-3z - 5a-3z3 + 9a-3z5 - 4a-3z7 - 2a-2 + 4a-2z2 - 10a-2z4 + 16a-2z6 - 7a-2z8 + a-1z-1 - 7a-1z3 + 7a-1z5 + 7a-1z7 - 6a-1z9 - 5 + 22z2 - 41z4 + 40z6 - 12z8 - 2z10 + 2az-1 - 7az + 9az3 - 14az5 + 22az7 - 12az9 - 3a2 + 16a2z2 - 35a2z4 + 34a2z6 - 12a2z8 - 2a2z10 - a3z5 + 5a3z7 - 6a3z9 + a4 - 2a4z2 - 2a4z4 + 8a4z6 - 7a4z8 - a5z-1 + 5a5z - 9a5z3 + 10a5z5 - 6a5z7 - a6z2 + 4a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

Khovanov Homology:

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 = 10                       1 j = 8                     3   j = 6                   6 1   j = 4                 8 3     j = 2               11 6       j = 0             11 8         j = -2           11 12           j = -4         8 10             j = -6       5 11               j = -8     3 8                 j = -10     5                   j = -12 1 3                     j = -14 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]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 214]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 214]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[22, 20, 7, 19], X[16, 5, 17, 6], > X[18, 10, 19, 9], X[4, 15, 5, 16], X[10, 18, 11, 17], X[14, 22, 15, 21], > X[20, 14, 21, 13], X[2, 7, 3, 8], X[6, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -6, 4, -11}, > {10, -1, 5, -7, 11, -2, 9, -8, 6, -4, 7, -5, 3, -9, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 8 13 19 21 22 -q + ----- - ---- + ---- - ---- + ---- - ------- + 19 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 14 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 4 2 4 4 -6 5 4 2 4 6 8 2 + q + --- - --- + --- + -- - q + -- - -- - 2 q - 3 q + 3 q - 3 q + 14 12 10 8 4 2 q q q q q q 10 12 14 > q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 214]][a, z]
Out[8]=	5 3 3 1 2 a a z z 3 5 z 4 z 3 --- - --- + -- + -- - - - a z - 4 a z + 2 a z + -- - ---- + 3 a z - a z z z 3 a 3 a a a 5 3 3 5 3 2 z 5 3 5 7 > 5 a z + a z - ---- + 3 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 214]][a, z]
Out[9]=	5 2 2 4 1 2 a a z 5 7 2 -5 - -- - 3 a + a + --- + --- - -- + -- - 7 a z + 5 a z - a z + 22 z - 2 a z z z 3 a a 2 2 3 3 z 4 z 2 2 4 2 6 2 5 z 7 z 3 5 3 > -- + ---- + 16 a z - 2 a z - a z - ---- - ---- + 9 a z - 9 a z + 4 2 3 a a a a 4 4 5 7 3 4 2 z 10 z 2 4 4 4 6 4 9 z > 2 a z - 41 z + ---- - ----- - 35 a z - 2 a z + 4 a z + ---- + 4 2 3 a a a 5 6 6 7 z 5 3 5 5 5 7 5 6 z 16 z 2 6 > ---- - 14 a z - a z + 10 a z - a z + 40 z - -- + ----- + 34 a z + a 4 2 a a 7 7 4 6 6 6 4 z 7 z 7 3 7 5 7 8 > 8 a z - 3 a z - ---- + ---- + 22 a z + 5 a z - 6 a z - 12 z - 3 a a 8 9 7 z 2 8 4 8 6 z 9 3 9 10 2 10 > ---- - 12 a z - 7 a z - ---- - 12 a z - 6 a z - 2 z - 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 1 3 5 3 8 5 11 11 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 8 10 11 2 2 2 4 2 4 3 > ----- + ---- + ---- + 8 t + 11 q t + 6 q t + 8 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a214

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