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L11a103 L11a105

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DrawMorseLink

PD Presentation:

X6172 X2,9,3,10 X12,3,13,4 X20,11,21,12 X18,6,19,5 X22,20,5,19 X10,21,11,22 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 8q-9/2 + 13q-7/2 - 19q-5/2 + 22q-3/2 - 23q-1/2 + 20q1/2 - 15q3/2 + 10q5/2 - 5q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 + q-16 + 5q-14 - 2q-12 + 3q-10 + 2q-8 - 4q-6 + 3q-4 - 4q-2 + 3 - 2q4 + 4q6 - 3q8 + q10 + 2q12 - q14

HOMFLY-PT Polynomial:

a-3z3 - a-1z-1 - 3a-1z - 4a-1z3 - 2a-1z5 + 3az-1 + 8az + 9az3 + 4az5 + az7 - 4a3z-1 - 9a3z - 6a3z3 - 2a3z5 + 2a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

a-4z4 - a-4z6 - a-3z - 3a-3z3 + 10a-3z5 - 5a-3z7 + a-2 - a-2z2 - 11a-2z4 + 21a-2z6 - 9a-2z8 - a-1z-1 + 2a-1z - 7a-1z3 + 8a-1z5 + 8a-1z7 - 7a-1z9 + 3 - 4z2 - 19z4 + 38z6 - 14z8 - 2z10 - 3az-1 + 13az - 24az3 + 11az5 + 15az7 - 12az9 + 3a2 - 4a2z2 - 14a2z4 + 27a2z6 - 12a2z8 - 2a2z10 - 4a3z-1 + 18a3z - 32a3z3 + 24a3z5 - 4a3z7 - 5a3z9 + 2a4 - 2a4z2 - 3a4z4 + 8a4z6 - 7a4z8 - 2a5z-1 + 7a5z - 10a5z3 + 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                     4   j = 6                   6 1   j = 4                 9 4     j = 2               11 6       j = 0             12 9         j = -2           11 12           j = -4         8 11             j = -6       5 11               j = -8     3 8                 j = -10   1 6                   j = -12   2                     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, 104]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 104]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[12, 3, 13, 4], X[20, 11, 21, 12], > X[18, 6, 19, 5], X[22, 20, 5, 19], X[10, 21, 11, 22], X[4, 17, 1, 18], > X[16, 14, 17, 13], X[14, 8, 15, 7], X[8, 16, 9, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -8}, {5, -1, 10, -11, 2, -7, 4, -3, 9, -10, 11, -9, 8, -5, > 6, -4, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 8 13 19 22 23 -q + ----- - ---- + ---- - ---- + ---- - ------- + 20 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 > 15 q + 10 q - 5 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -16 5 2 3 2 4 3 4 4 6 8 3 + q + q + --- - --- + --- + -- - -- + -- - -- - 2 q + 4 q - 3 q + 14 12 10 8 6 4 2 q q q q q q q 10 12 14 > q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 104]][a, z]
Out[8]=	3 5 3 3 1 3 a 4 a 2 a 3 z 3 5 z 4 z -(---) + --- - ---- + ---- - --- + 8 a z - 9 a z + 2 a z + -- - ---- + a z z z z a 3 a a 5 3 3 3 5 3 2 z 5 3 5 7 > 9 a z - 6 a z + a z - ---- + 4 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 104]][a, z]
Out[9]=	3 5 -2 2 4 1 3 a 4 a 2 a z 2 z 3 3 + a + 3 a + 2 a - --- - --- - ---- - ---- - -- + --- + 13 a z + 18 a z + a z z z z 3 a a 2 3 3 5 7 2 z 2 2 4 2 6 2 3 z 7 z > 7 a z - a z - 4 z - -- - 4 a z - 2 a z - a z - ---- - ---- - 2 3 a a a 4 4 3 3 3 5 3 7 3 4 z 11 z 2 4 > 24 a z - 32 a z - 10 a z + 2 a z - 19 z + -- - ----- - 14 a z - 4 2 a a 5 5 4 4 6 4 10 z 8 z 5 3 5 5 5 7 5 > 3 a z + 4 a z + ----- + ---- + 11 a z + 24 a z + 10 a z - a z + 3 a a 6 6 7 7 6 z 21 z 2 6 4 6 6 6 5 z 8 z 7 > 38 z - -- + ----- + 27 a z + 8 a z - 3 a z - ---- + ---- + 15 a z - 4 2 3 a a a a 8 9 3 7 5 7 8 9 z 2 8 4 8 7 z 9 > 4 a z - 6 a z - 14 z - ---- - 12 a z - 7 a z - ---- - 12 a z - 2 a a 3 9 10 2 10 > 5 a z - 2 z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 2 1 6 3 8 5 11 12 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 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 11 11 2 2 2 4 2 4 3 > ----- + ---- + ---- + 9 t + 11 q t + 6 q t + 9 q t + 4 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 + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a104

L11a103 L11a105