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L11a154 L11a156

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 6q-11/2 - 11q-9/2 + 14q-7/2 - 17q-5/2 + 17q-3/2 - 15q-1/2 + 11q1/2 - 7q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 + 2q-20 - 2q-18 + 2q-16 + 4q-14 - 2q-12 + 3q-10 - q-8 - q-6 + q-4 - 2q-2 + 5 - q2 + 3q6 - 2q8 + q12

HOMFLY-PT Polynomial:

- a-3z - a-1z-1 + 2a-1z3 + 2az-1 + 2az + az3 - az5 - 2a3z-1 - 2a3z - a3z5 + a5z-1 + a5z + 2a5z3 - a7z

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 3a-2z6 - a-1z-1 + 3a-1z - 5a-1z3 + 8a-1z5 - 5a-1z7 + z2 - z4 + 5z6 - 5z8 - 2az-1 + 10az - 18az3 + 16az5 - 4az7 - 3az9 - a2 + 4a2z2 - 13a2z4 + 17a2z6 - 8a2z8 - a2z10 - 2a3z-1 + 10a3z - 18a3z3 + 10a3z5 + 5a3z7 - 6a3z9 + 3a4z2 - 14a4z4 + 19a4z6 - 7a4z8 - a4z10 - a5z-1 + 7a5z - 15a5z3 + 12a5z5 + a5z7 - 3a5z9 - 4a6z4 + 9a6z6 - 4a6z8 + 3a7z - 8a7z3 + 9a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

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 = 8                       1 j = 6                     2   j = 4                   5 1   j = 2                 6 2     j = 0               9 5       j = -2             9 7         j = -4           8 8           j = -6         7 10             j = -8       4 7               j = -10     2 7                 j = -12   1 4                   j = -14   2                     j = -16 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, 155]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 155]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 6, 15, 5], X[20, 11, 21, 12], > X[22, 18, 7, 17], X[18, 22, 19, 21], X[16, 13, 17, 14], X[12, 19, 13, 20], > X[4, 16, 5, 15], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 3, -11}, > {10, -1, 11, -2, 4, -8, 7, -3, 9, -7, 5, -6, 8, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 6 11 14 17 17 15 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 11 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 7 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 2 2 4 2 3 -8 -6 -4 2 2 5 - q + --- - --- + --- + --- - --- + --- - q - q + q - -- - q + 20 18 16 14 12 10 2 q q q q q q q 6 8 12 > 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 155]][a, z]
Out[8]=	3 5 3 1 2 a 2 a a z 3 5 7 2 z 3 -(---) + --- - ---- + -- - -- + 2 a z - 2 a z + a z - a z + ---- + a z + a z z z z 3 a a 5 3 5 3 5 > 2 a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 155]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a z 3 z 3 5 7 -a - --- - --- - ---- - -- - -- + --- + 10 a z + 10 a z + 7 a z + 3 a z + a z z z z 3 a a 2 3 3 2 2 z 2 2 4 2 8 2 2 z 5 z 3 > z - ---- + 4 a z + 3 a z - 2 a z + ---- - ---- - 18 a z - 2 3 a a a 4 3 3 5 3 7 3 4 5 z 2 4 4 4 6 4 > 18 a z - 15 a z - 8 a z - z + ---- - 13 a z - 14 a z - 4 a z + 2 a 5 5 8 4 z 8 z 5 3 5 5 5 7 5 6 > 3 a z - -- + ---- + 16 a z + 10 a z + 12 a z + 9 a z + 5 z - 3 a a 6 7 3 z 2 6 4 6 6 6 8 6 5 z 7 3 7 > ---- + 17 a z + 19 a z + 9 a z - a z - ---- - 4 a z + 5 a z + 2 a a 5 7 7 7 8 2 8 4 8 6 8 9 3 9 > a z - 3 a z - 5 z - 8 a z - 7 a z - 4 a z - 3 a z - 6 a z - 5 9 2 10 4 10 > 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 2 1 4 2 7 4 7 9 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 7 10 8 8 9 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 5 t + 6 q t + 2 q t + 5 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a155

L11a154 L11a156