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L11n442 L11n444

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DrawMorseLink

PD Presentation:

X6172 X2536 X18,11,19,12 X3,11,4,10 X9,1,10,4 X7,17,8,16 X15,5,16,8 X20,14,21,13 X22,19,15,20 X12,22,13,21 X14,17,9,18

Gauss Code:

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

Jones Polynomial:

q-9/2 - 2q-7/2 + 2q-5/2 - 3q-3/2 + q-1/2 - 3q1/2 - q3/2 - q5/2 - 2q7/2 + q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-14 - q-10 + 2q-6 + 3q-4 + 7q-2 + 8 + 11q2 + 12q4 + 11q6 + 11q8 + 7q10 + 5q12 + 3q14 + 2q16 + q18

HOMFLY-PT Polynomial:

- a-5z-3 - 2a-5z-1 - a-5z + 3a-3z-3 + 8a-3z-1 + 8a-3z + 2a-3z3 - 3a-1z-3 - 11a-1z-1 - 14a-1z - 7a-1z3 - a-1z5 + az-3 + 6az-1 + 9az + 5az3 + az5 - a3z-1 - 2a3z - a3z3

Kauffman Polynomial:

a-5z-3 - 5a-5z-1 + 10a-5z - 11a-5z3 + 6a-5z5 - a-5z7 - 3a-4z-2 + 9a-4 - 7a-4z2 - 3a-4z4 + 5a-4z6 - a-4z8 + 3a-3z-3 - 14a-3z-1 + 35a-3z - 42a-3z3 + 21a-3z5 - 3a-3z7 - 6a-2z-2 + 21a-2 - 24a-2z2 + 3a-2z4 + 5a-2z6 - a-2z8 + 3a-1z-3 - 18a-1z-1 + 49a-1z - 60a-1z3 + 29a-1z5 - 4a-1z7 - 3z-2 + 18 - 28z2 + 10z4 + 3z6 - z8 + az-3 - 11az-1 + 30az - 39az3 + 23az5 - 4az7 + 6a2 - 14a2z2 + 8a2z4 + 2a2z6 - a2z8 - 2a3z-1 + 6a3z - 10a3z3 + 9a3z5 - 2a3z7 + a4 - 3a4z2 + 4a4z4 - 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 = 12                       1 j = 10                         j = 8                   2 1   j = 6               3 1 1     j = 4             1 4 1       j = 2           4 2 2         j = 0         2 5 1           j = -2       2 1 1             j = -4     1 3 1               j = -6   1 1                   j = -8   1                     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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 443]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 443]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 11, 19, 12], X[3, 11, 4, 10], > X[9, 1, 10, 4], X[7, 17, 8, 16], X[15, 5, 16, 8], X[20, 14, 21, 13], > X[22, 19, 15, 20], X[12, 22, 13, 21], X[14, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -10, 8, -11}, > {-7, 6, 11, -3, 9, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 2 2 3 1 3/2 5/2 7/2 q - ---- + ---- - ---- + ------- - 3 Sqrt[q] - q - q - 2 q + 7/2 5/2 3/2 Sqrt[q] q q q 9/2 11/2 > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -10 2 3 7 2 4 6 8 10 8 - q - q + -- + -- + -- + 11 q + 12 q + 11 q + 11 q + 7 q + 6 4 2 q q q 12 14 16 18 > 5 q + 3 q + 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 443]][a, z]
Out[8]=	3 1 3 3 a 2 8 11 6 a a z 8 z 14 z -(-----) + ----- - ---- + -- - ---- + ---- - --- + --- - -- - -- + --- - ---- + 5 3 3 3 3 3 5 3 a z z z 5 3 a a z a z a z z a z a z a a 3 3 5 3 2 z 7 z 3 3 3 z 5 > 9 a z - 2 a z + ---- - ---- + 5 a z - a z - -- + a z 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 443]][a, z]
Out[9]=	9 21 2 4 1 3 3 a 3 3 6 18 + -- + -- + 6 a + a + ----- + ----- + ---- + -- - -- - ----- - ----- - 4 2 5 3 3 3 3 3 2 4 2 2 2 a a a z a z a z z z a z a z 3 5 14 18 11 a 2 a 10 z 35 z 49 z 3 > ---- - ---- - --- - ---- - ---- + ---- + ---- + ---- + 30 a z + 6 a z - 5 3 a z z z 5 3 a a z a z a a 2 2 3 3 3 2 7 z 24 z 2 2 4 2 11 z 42 z 60 z > 28 z - ---- - ----- - 14 a z - 3 a z - ----- - ----- - ----- - 4 2 5 3 a a a a a 4 4 5 3 3 3 4 3 z 3 z 2 4 4 4 6 z > 39 a z - 10 a z + 10 z - ---- + ---- + 8 a z + 4 a z + ---- + 4 2 5 a a a 5 5 6 6 21 z 29 z 5 3 5 6 5 z 5 z 2 6 4 6 > ----- + ----- + 23 a z + 9 a z + 3 z + ---- + ---- + 2 a z - a z - 3 a 4 2 a a a 7 7 7 8 8 z 3 z 4 z 7 3 7 8 z z 2 8 > -- - ---- - ---- - 4 a z - 2 a z - z - -- - -- - a z 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 1 1 1 3 2 2 5 + q + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 1 1 2 4 2 2 4 2 6 2 4 3 > ---- + ---- + t + 2 q t + q t + 2 q t + 4 q t + 3 q t + q t + 4 2 q t q t 6 3 6 4 8 4 8 5 12 6 > q t + q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n443

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