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L11a145 L11a147

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X16,8,17,7 X20,14,21,13 X22,16,7,15 X6,19,1,20 X18,11,19,12 X12,6,13,5 X14,22,15,21 X4,18,5,17

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 5q-5/2 - 9q-3/2 + 11q-1/2 - 14q1/2 + 13q3/2 - 12q5/2 + 9q7/2 - 5q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-12 + q-10 - q-8 + 3q-6 + 2q-4 + 2q-2 + 5 - q2 + 4q4 - 2q6 - q10 - 3q12 + q14 - q16 + q18

HOMFLY-PT Polynomial:

a-3z-1 - a-3z - 7a-3z3 - 5a-3z5 - a-3z7 - 3a-1z-1 + a-1z + 15a-1z3 + 17a-1z5 + 7a-1z7 + a-1z9 + 2az-1 - az - 7az3 - 5az5 - az7

Kauffman Polynomial:

2a-7z3 - a-7z5 - 3a-6z2 + 7a-6z4 - 3a-6z6 - 3a-5z3 + 8a-5z5 - 4a-5z7 + a-4 - 3a-4z2 - 2a-4z4 + 7a-4z6 - 4a-4z8 - a-3z-1 - 2a-3z + 13a-3z3 - 20a-3z5 + 12a-3z7 - 4a-3z9 + 3a-2 - 7a-2z4 + 3a-2z8 - 2a-2z10 - 3a-1z-1 - a-1z + 31a-1z3 - 53a-1z5 + 32a-1z7 - 8a-1z9 + 3 + z2 - 8z4 + 3z6 + 3z8 - 2z10 - 2az-1 + 2az + 6az3 - 14az5 + 13az7 - 4az9 - 7a2z4 + 12a2z6 - 4a2z8 + a3z - 7a3z3 + 10a3z5 - 3a3z7 - a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     2   j = 10                   3 1   j = 8                 6 2     j = 6               6 3       j = 4             7 6         j = 2           7 6           j = 0         5 8             j = -2       4 6               j = -4     2 6                 j = -6   1 3                   j = -8   2                     j = -10 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, 146]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 146]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 8, 17, 7], > X[20, 14, 21, 13], X[22, 16, 7, 15], X[6, 19, 1, 20], X[18, 11, 19, 12], > X[12, 6, 13, 5], X[14, 22, 15, 21], X[4, 18, 5, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 9, -7}, > {4, -1, 2, -3, 8, -9, 5, -10, 6, -4, 11, -8, 7, -5, 10, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 5 9 11 3/2 5/2 q - ---- + ---- - ---- + ------- - 14 Sqrt[q] + 13 q - 12 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 9 q - 5 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 -8 3 2 2 2 4 6 10 12 14 5 - q + q - q + -- + -- + -- - q + 4 q - 2 q - q - 3 q + q - 6 4 2 q q q 16 18 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 146]][a, z]
Out[8]=	3 3 5 5 1 3 2 a z z 7 z 15 z 3 5 z 17 z ---- - --- + --- - -- + - - a z - ---- + ----- - 7 a z - ---- + ----- - 3 a z z 3 a 3 a 3 a a z a a a 7 7 9 5 z 7 z 7 z > 5 a z - -- + ---- - a z + -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 146]][a, z]
Out[9]=	2 2 -4 3 1 3 2 a 2 z z 3 2 3 z 3 z 3 + a + -- - ---- - --- - --- - --- - - + 2 a z + a z + z - ---- - ---- - 2 3 a z z 3 a 6 4 a a z a a a 3 3 3 3 4 4 2 2 z 3 z 13 z 31 z 3 3 3 4 7 z > a z + ---- - ---- + ----- + ----- + 6 a z - 7 a z - 8 z + ---- - 7 5 3 a 6 a a a a 4 4 5 5 5 5 2 z 7 z 2 4 4 4 z 8 z 20 z 53 z 5 > ---- - ---- - 7 a z + 3 a z - -- + ---- - ----- - ----- - 14 a z + 4 2 7 5 3 a a a a a a 6 6 7 7 7 3 5 6 3 z 7 z 2 6 4 6 4 z 12 z 32 z > 10 a z + 3 z - ---- + ---- + 12 a z - a z - ---- + ----- + ----- + 6 4 5 3 a a a a a 8 8 9 9 7 3 7 8 4 z 3 z 2 8 4 z 8 z 9 > 13 a z - 3 a z + 3 z - ---- + ---- - 4 a z - ---- - ---- - 4 a z - 4 2 3 a a a a 10 10 2 z > 2 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 3 2 6 4 5 6 8 + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 6 q t + 7 q t + 6 q t + 6 q t + 3 q t + 6 q t + 2 q t + 10 4 10 5 12 5 14 6 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a146

L11a145 L11a147