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L11n310 L11n312

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,20,10,21 X15,5,16,10 X19,15,20,14 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-5 - q-4 + 2q-3 + q-2 + 3 - 3q + 3q2 - 3q3 + 2q4 - q5

A2 (sl(3)) Invariant:

q-16 + 2q-14 + 3q-12 + 4q-10 + 6q-8 + 7q-6 + 4q-4 + 5q-2 + 2 - q4 - 3q6 - q8 - 2q10 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 - a-2z-2 - 4a-2 - 3a-2z2 + 4z-2 + 13 + 14z2 + 7z4 + z6 - 5a2z-2 - 12a2 - 10a2z2 - 2a2z4 + 2a4z-2 + 3a4 + a4z2

Kauffman Polynomial:

a-5z - 3a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 2a-4z6 + a-3z-1 - 4a-3z + 6a-3z3 - 7a-3z5 + 2a-3z7 - a-2z-2 + 5a-2 - 8a-2z2 + 6a-2z4 - 4a-2z6 + a-2z8 + 5a-1z-1 - 21a-1z + 33a-1z3 - 18a-1z5 + 3a-1z7 - 4z-2 + 18 - 42z2 + 49z4 - 22z6 + 3z8 + 9az-1 - 29az + 30az3 - 3az5 - 5az7 + az9 - 5a2z-2 + 21a2 - 49a2z2 + 53a2z4 - 23a2z6 + 3a2z8 + 5a3z-1 - 13a3z + 6a3z3 + 7a3z5 - 6a3z7 + a3z9 - 2a4z-2 + 9a4 - 17a4z2 + 16a4z4 - 7a4z6 + a4z8

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 = 11                       1 j = 9                     1   j = 7                   2 1   j = 5               1 2 1     j = 3             1 3 2       j = 1           1 3 2         j = -1         1 2 4           j = -3       1 3 1             j = -5     1 1 3               j = -7   1 2                   j = -9                         j = -11 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 311]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 311]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[13, 19, 14, 18], X[17, 11, 18, 22], > X[7, 17, 8, 16], X[21, 8, 22, 9], X[9, 20, 10, 21], X[15, 5, 16, 10], > X[19, 15, 20, 14], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -5, 6, -7, 8}, > {11, -2, -3, 9, -8, 5, -4, 3, -9, 7, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -4 2 -2 2 3 4 5 3 + q - q + -- + q - 3 q + 3 q - 3 q + 2 q - q 3 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 3 4 6 7 4 5 4 6 8 10 14 2 + q + --- + --- + --- + -- + -- + -- + -- - q - 3 q - q - 2 q + q - 14 12 10 8 6 4 2 q q q q q q q 16 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 311]][a, z]
Out[8]=	2 4 2 2 4 2 4 4 1 5 a 2 a 2 z 3 z 13 - -- - 12 a + 3 a + -- - ----- - ---- + ---- + 14 z - -- - ---- - 2 2 2 2 2 2 4 2 a z a z z z a a 2 2 4 2 4 2 4 6 > 10 a z + a z + 7 z - 2 a z + z
In[9]:=	Kauffman[Link[11, NonAlternating, 311]][a, z]
Out[9]=	2 4 3 5 2 4 4 1 5 a 2 a 1 5 9 a 5 a 18 + -- + 21 a + 9 a - -- - ----- - ---- - ---- + ---- + --- + --- + ---- + 2 2 2 2 2 2 3 a z z z a z a z z z a z 2 2 z 4 z 21 z 3 2 2 z 8 z 2 2 > -- - --- - ---- - 29 a z - 13 a z - 42 z + ---- - ---- - 49 a z - 5 3 a 4 2 a a a a 3 3 3 4 4 4 2 3 z 6 z 33 z 3 3 3 4 6 z 6 z > 17 a z - ---- + ---- + ----- + 30 a z + 6 a z + 49 z - ---- + ---- + 5 3 a 4 2 a a a a 5 5 5 6 2 4 4 4 z 7 z 18 z 5 3 5 6 2 z > 53 a z + 16 a z + -- - ---- - ----- - 3 a z + 7 a z - 22 z + ---- - 5 3 a 4 a a a 6 7 7 8 4 z 2 6 4 6 2 z 3 z 7 3 7 8 z > ---- - 23 a z - 7 a z + ---- + ---- - 5 a z - 6 a z + 3 z + -- + 2 3 a 2 a a a 2 8 4 8 9 3 9 > 3 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 3 1 1 2 1 1 1 3 3 - + 3 q + q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + q 11 6 7 5 7 4 5 4 5 3 3 3 5 2 3 2 q t q t q t q t q t q t q t q t 1 1 2 q 3 5 3 2 5 2 5 3 > ---- + ---- + --- + - + 2 q t + 3 q t + q t + 2 q t + 2 q t + q t + 2 3 q t t q t q t 7 3 7 4 9 4 11 5 > 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n311

L11n310 L11n312