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L11a400 L11a402

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DrawMorseLink

PD Presentation:

X6172 X18,12,19,11 X8493 X2,16,3,15 X16,7,17,8 X22,9,11,10 X4,17,1,18 X10,19,5,20 X12,6,13,5 X14,21,15,22 X20,13,21,14

Gauss Code:

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

Jones Polynomial:

- q-8 + 5q-7 - 12q-6 + 20q-5 - 27q-4 + 32q-3 - 30q-2 + 28q-1 - 19 + 12q - 5q2 + q3

A2 (sl(3)) Invariant:

- q-24 + 2q-22 - 3q-18 + 5q-16 - 5q-14 + 3q-12 + 4q-10 + q-8 + 12q-6 - q-4 + 9q-2 + 1 - 2q2 + 4q4 - 3q6 + q8

HOMFLY-PT Polynomial:

z-2 + 1 + z2 + 2z4 + z6 - 2a2z-2 - a2 - 3a2z2 - 6a2z4 - 4a2z6 - a2z8 + a4z-2 + 3a4z2 + 5a4z4 + 2a4z6 - a6z2 - a6z4

Kauffman Polynomial:

- a-2z4 + a-2z6 + 3a-1z3 - 8a-1z5 + 5a-1z7 - z-2 + 1 - 4z2 + 16z4 - 23z6 + 11z8 + 2az-1 - az + 6az3 - 4az5 - 14az7 + 11az9 - 2a2z-2 + a2 - 12a2z2 + 51a2z4 - 67a2z6 + 22a2z8 + 4a2z10 + 2a3z-1 - a3z + 8a3z3 + 2a3z5 - 37a3z7 + 24a3z9 - a4z-2 + a4 - 12a4z2 + 51a4z4 - 73a4z6 + 28a4z8 + 4a4z10 + 10a5z3 - 17a5z5 - 6a5z7 + 13a5z9 - 4a6z2 + 14a6z4 - 25a6z6 + 17a6z8 + 5a7z3 - 14a7z5 + 12a7z7 - 3a8z4 + 5a8z6 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     4   j = 3                   8 1   j = 1                 11 4     j = -1               17 8       j = -3             15 13         j = -5           17 15           j = -7         12 17             j = -9       8 15               j = -11     4 12                 j = -13   1 8                   j = -15   4                     j = -17 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, 401]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 401]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 12, 19, 11], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[22, 9, 11, 10], X[4, 17, 1, 18], X[10, 19, 5, 20], > X[12, 6, 13, 5], X[14, 21, 15, 22], X[20, 13, 21, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -7}, {9, -1, 5, -3, 6, -8}, > {2, -9, 11, -10, 4, -5, 7, -2, 8, -11, 10, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-8 5 12 20 27 32 30 28 2 3 -19 - q + -- - -- + -- - -- + -- - -- + -- + 12 q - 5 q + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 3 5 5 3 4 -8 12 -4 9 2 1 - q + --- - --- + --- - --- + --- + --- + q + -- - q + -- - 2 q + 22 18 16 14 12 10 6 2 q q q q q q q q 4 6 8 > 4 q - 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 401]][a, z]
Out[8]=	2 4 2 -2 2 a a 2 2 2 4 2 6 2 4 2 4 1 - a + z - ---- + -- + z - 3 a z + 3 a z - a z + 2 z - 6 a z + 2 2 z z 4 4 6 4 6 2 6 4 6 2 8 > 5 a z - a z + z - 4 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 401]][a, z]
Out[9]=	2 4 3 2 4 -2 2 a a 2 a 2 a 3 2 2 2 1 + a + a - z - ---- - -- + --- + ---- - a z - a z - 4 z - 12 a z - 2 2 z z z z 3 4 2 6 2 3 z 3 3 3 5 3 7 3 4 > 12 a z - 4 a z + ---- + 6 a z + 8 a z + 10 a z + 5 a z + 16 z - a 4 5 z 2 4 4 4 6 4 8 4 8 z 5 3 5 > -- + 51 a z + 51 a z + 14 a z - 3 a z - ---- - 4 a z + 2 a z - 2 a a 6 5 5 7 5 9 5 6 z 2 6 4 6 6 6 > 17 a z - 14 a z + a z - 23 z + -- - 67 a z - 73 a z - 25 a z + 2 a 7 8 6 5 z 7 3 7 5 7 7 7 8 > 5 a z + ---- - 14 a z - 37 a z - 6 a z + 12 a z + 11 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 22 a z + 28 a z + 17 a z + 11 a z + 24 a z + 13 a z + 4 a z + 4 10 > 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	13 17 1 4 1 8 4 12 8 15 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 12 17 17 15 15 8 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 11 q t + 4 q t + 8 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a401

L11a400 L11a402