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L11a410 L11a412

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X22,16,11,15 X14,8,15,7 X10,12,5,11 X20,18,21,17 X8,19,9,20 X18,9,19,10 X16,22,17,21 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 6q-1 - 8 + 13q - 14q2 + 15q3 - 13q4 + 10q5 - 6q6 + 3q7 - q8

A2 (sl(3)) Invariant:

q-10 + q-8 + q-6 + 5q-4 + 3q-2 + 4 + 6q2 + 4q6 + q10 + 2q12 - 2q14 + 3q16 - q18 - q20 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + 2a-4 + 4a-4z2 + 3a-4z4 + a-4z6 + a-2z-2 + a-2 + a-2z2 + 2a-2z4 + a-2z6 - 2z-2 - 4 - 5z2 - 2z4 + a2z-2 + 2a2 + a2z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 3a-8z6 - 2a-7z + 7a-7z3 - 11a-7z5 + 5a-7z7 + 2a-6 - 9a-6z2 + 19a-6z4 - 16a-6z6 + 6a-6z8 - 4a-5z + 8a-5z3 + 3a-5z5 - 7a-5z7 + 4a-5z9 + 4a-4 - 22a-4z2 + 41a-4z4 - 27a-4z6 + 7a-4z8 + a-4z10 - 8a-3z3 + 18a-3z5 - 15a-3z7 + 6a-3z9 + a-2z-2 - 2a-2 - 3a-2z2 + 7a-2z4 - 9a-2z6 + 3a-2z8 + a-2z10 - 2a-1z-1 + 6a-1z - 6a-1z3 - 2a-1z5 - a-1z7 + 2a-1z9 + 2z-2 - 7 + 15z2 - 13z4 + 2z8 - 2az-1 + 4az + az3 - 5az5 + 2az7 + a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6

Khovanov Homology:

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                     2   j = 13                   4 1   j = 11                 6 2     j = 9               7 4       j = 7             8 6         j = 5           7 8           j = 3         6 7             j = 1       4 9               j = -1     2 4                 j = -3   1 5                   j = -5   1                     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]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 411]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 411]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[22, 16, 11, 15], X[14, 8, 15, 7], > X[10, 12, 5, 11], X[20, 18, 21, 17], X[8, 19, 9, 20], X[18, 9, 19, 10], > X[16, 22, 17, 21], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -7, 8, -5}, > {5, -2, 11, -4, 3, -9, 6, -8, 7, -6, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 6 2 3 4 5 6 7 8 -8 + q - -- + - + 13 q - 14 q + 15 q - 13 q + 10 q - 6 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 5 3 2 6 10 12 14 16 4 + q + q + q + -- + -- + 6 q + 4 q + q + 2 q - 2 q + 3 q - 4 2 q q 18 20 22 24 > q - q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 411]][a, z]
Out[8]=	2 2 2 2 -6 2 -2 2 2 1 a 2 2 z 4 z z -4 - a + -- + a + 2 a - -- + ----- + -- - 5 z - ---- + ---- + -- + 4 2 2 2 2 6 4 2 a z a z z a a a 4 4 4 6 6 2 2 4 z 3 z 2 z z z > a z - 2 z - -- + ---- + ---- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 411]][a, z]
Out[9]=	2 2 4 2 2 2 1 a 2 2 a 2 z 4 z 6 z -7 + -- + -- - -- - 4 a + -- + ----- + -- - --- - --- - --- - --- + --- + 6 4 2 2 2 2 2 a z z 7 5 a a a a z a z z a a 2 2 2 2 3 3 3 2 z 9 z 22 z 3 z 2 2 2 z 7 z 8 z > 4 a z + 15 z + -- - ---- - ----- - ---- + 6 a z - ---- + ---- + ---- - 8 6 4 2 9 7 5 a a a a a a a 3 3 4 4 4 4 5 8 z 6 z 3 4 6 z 19 z 41 z 7 z 2 4 z > ---- - ---- + a z - 13 z - ---- + ----- + ----- + ---- - 4 a z + -- - 3 a 8 6 4 2 9 a a a a a a 5 5 5 5 6 6 6 6 11 z 3 z 18 z 2 z 5 3 z 16 z 27 z 9 z > ----- + ---- + ----- - ---- - 5 a z + ---- - ----- - ----- - ---- + 7 5 3 a 8 6 4 2 a a a a a a a 7 7 7 7 8 8 8 2 6 5 z 7 z 15 z z 7 8 6 z 7 z 3 z > a z + ---- - ---- - ----- - -- + 2 a z + 2 z + ---- + ---- + ---- + 7 5 3 a 6 4 2 a a a a a a 9 9 9 10 10 4 z 6 z 2 z z z > ---- + ---- + ---- + --- + --- 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 5 2 4 4 q 3 9 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 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 > 7 q t + 8 q t + 8 q t + 6 q t + 7 q t + 4 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a411

L11a410 L11a412