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L11a134 L11a136

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 4q-17/2 - 9q-15/2 + 15q-13/2 - 20q-11/2 + 22q-9/2 - 23q-7/2 + 19q-5/2 - 14q-3/2 + 8q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - 2q-26 + 2q-24 - 3q-20 + 4q-18 - 3q-16 + 4q-14 + 2q-12 + 5q-8 - 4q-6 + 3q-4 - 1 + 2q2 - q4

HOMFLY-PT Polynomial:

2az3 + az5 - a3z-1 - 2a3z - 3a3z3 - 3a3z5 - a3z7 + a5z-1 - a5z - 3a5z3 - 3a5z5 - a5z7 + a7z + 2a7z3 + a7z5

Kauffman Polynomial:

2z4 - z6 - 6az3 + 10az5 - 4az7 + 2a2z2 - 14a2z4 + 18a2z6 - 7a2z8 - a3z-1 + 2a3z - 4a3z3 + 9a3z7 - 6a3z9 + a4 + 4a4z2 - 24a4z4 + 32a4z6 - 11a4z8 - 2a4z10 - a5z-1 + a5z + 4a5z3 - 9a5z5 + 19a5z7 - 12a5z9 + 3a6z2 - 14a6z4 + 27a6z6 - 13a6z8 - 2a6z10 - 6a7z3 + 14a7z5 - 2a7z7 - 6a7z9 - a8z2 - a8z4 + 10a8z6 - 9a8z8 + a9z - 7a9z3 + 12a9z5 - 8a9z7 - 2a10z2 + 5a10z4 - 4a10z6 + a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     3   j = 0                   5 1   j = -2                 9 3     j = -4               11 6       j = -6             12 8         j = -8           10 11           j = -10         10 12             j = -12       6 11               j = -14     3 9                 j = -16   1 6                   j = -18   3                     j = -20 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, 135]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 135]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[14, 3, 15, 4], X[10, 5, 11, 6], > X[20, 11, 21, 12], X[22, 13, 5, 14], X[12, 21, 13, 22], X[4, 19, 1, 20], > X[18, 16, 19, 15], X[16, 8, 17, 7], X[8, 18, 9, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -8}, {4, -1, 10, -11, 2, -4, 5, -7, 6, -3, 9, -10, 11, -9, > 8, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 4 9 15 20 22 23 19 14 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 8 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 2 3 4 3 4 2 5 4 3 2 4 -1 + q - --- + --- - --- + --- - --- + --- + --- + -- - -- + -- + 2 q - q 26 24 20 18 16 14 12 8 6 4 q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 135]][a, z]
Out[8]=	3 5 a a 3 5 7 3 3 3 5 3 7 3 -(--) + -- - 2 a z - a z + a z + 2 a z - 3 a z - 3 a z + 2 a z + z z 5 3 5 5 5 7 5 3 7 5 7 > a z - 3 a z - 3 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 135]][a, z]
Out[9]=	3 5 4 a a 3 5 9 2 2 4 2 6 2 8 2 a - -- - -- + 2 a z + a z + a z + 2 a z + 4 a z + 3 a z - a z - z z 10 2 3 3 3 5 3 7 3 9 3 11 3 4 > 2 a z - 6 a z - 4 a z + 4 a z - 6 a z - 7 a z + a z + 2 z - 2 4 4 4 6 4 8 4 10 4 5 5 5 > 14 a z - 24 a z - 14 a z - a z + 5 a z + 10 a z - 9 a z + 7 5 9 5 11 5 6 2 6 4 6 6 6 > 14 a z + 12 a z - a z - z + 18 a z + 32 a z + 27 a z + 8 6 10 6 7 3 7 5 7 7 7 9 7 > 10 a z - 4 a z - 4 a z + 9 a z + 19 a z - 2 a z - 8 a z - 2 8 4 8 6 8 8 8 3 9 5 9 7 9 > 7 a z - 11 a z - 13 a z - 9 a z - 6 a z - 12 a z - 6 a z - 4 10 6 10 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 9 1 3 1 6 3 9 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 11 10 12 10 11 12 8 11 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 5 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 3 t 2 2 2 4 3 > --- + t + 3 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a135

L11a134 L11a136