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L11n413 L11n415

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DrawMorseLink

PD Presentation:

X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X15,20,16,21 X19,22,20,13 X21,16,22,17

Gauss Code:

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

Jones Polynomial:

- q-7 + q-6 - q-5 + q-4 + q-3 + 2q-1 - 1 + 2q - q2 + q3

A2 (sl(3)) Invariant:

- q-22 - q-20 - 2q-18 - 2q-16 - q-14 + 4q-10 + 5q-8 + 7q-6 + 6q-4 + 4q-2 + 3 + q2 + q4 + q6 + q8 + q10

HOMFLY-PT Polynomial:

2a-2 + a-2z2 + 2z-2 - 3z2 - z4 - 5a2z-2 - 8a2 - 5a2z2 - a2z4 + 4a4z-2 + 8a4 + 5a4z2 + a4z4 - a6z-2 - 2a6 - a6z2

Kauffman Polynomial:

- 2a-2 + 6a-2z2 - 5a-2z4 + a-2z6 + a-1z + 2a-1z3 - 4a-1z5 + a-1z7 - 2z-2 + 6 - 7z2 + 7z4 - 5z6 + z8 + 5az-1 - 16az + 21az3 - 12az5 + 2az7 - 5a2z-2 + 20a2 - 31a2z2 + 22a2z4 - 8a2z6 + a2z8 + 9a3z-1 - 33a3z + 37a3z3 - 16a3z5 + 2a3z7 - 4a4z-2 + 17a4 - 25a4z2 + 20a4z4 - 8a4z6 + a4z8 + 5a5z-1 - 21a5z + 28a5z3 - 14a5z5 + 2a5z7 - a6z-2 + 4a6 - 7a6z2 + 10a6z4 - 6a6z6 + a6z8 + a7z-1 - 5a7z + 10a7z3 - 6a7z5 + a7z7

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 = 7                       1 j = 5                     1 1 j = 3                   1     j = 1               1 1 1     j = -1             1 3 1       j = -3           1   1         j = -5         1 4 2           j = -7       1   1             j = -9       1 1               j = -11   1 1                   j = -13                         j = -15 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, 414]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 414]]
Out[4]=	PD[X[8, 1, 9, 2], X[5, 15, 6, 14], X[10, 3, 11, 4], X[13, 5, 14, 4], > X[2, 7, 3, 8], X[6, 9, 1, 10], X[18, 12, 19, 11], X[12, 18, 7, 17], > X[15, 20, 16, 21], X[19, 22, 20, 13], X[21, 16, 22, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, 7, -8}, > {-4, 2, -9, 11, 8, -7, -10, 9, -11, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-7 -6 -5 -4 -3 2 2 3 -1 - q + q - q + q + q + - + 2 q - q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 2 2 -14 4 5 7 6 4 2 4 6 3 - q - q - --- - --- - q + --- + -- + -- + -- + -- + q + q + q + 18 16 10 8 6 4 2 q q q q q q q 8 10 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 414]][a, z]
Out[8]=	2 4 6 2 2 2 4 6 2 5 a 4 a a 2 z 2 2 -- - 8 a + 8 a - 2 a + -- - ---- + ---- - -- - 3 z + -- - 5 a z + 2 2 2 2 2 2 a z z z z a 4 2 6 2 4 2 4 4 4 > 5 a z - a z - z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 414]][a, z]
Out[9]=	2 4 6 3 5 2 2 4 6 2 5 a 4 a a 5 a 9 a 5 a 6 - -- + 20 a + 17 a + 4 a - -- - ---- - ---- - -- + --- + ---- + ---- + 2 2 2 2 2 z z z a z z z z 7 2 a z 3 5 7 2 6 z 2 2 > -- + - - 16 a z - 33 a z - 21 a z - 5 a z - 7 z + ---- - 31 a z - z a 2 a 3 4 2 6 2 2 z 3 3 3 5 3 7 3 > 25 a z - 7 a z + ---- + 21 a z + 37 a z + 28 a z + 10 a z + a 4 5 4 5 z 2 4 4 4 6 4 4 z 5 3 5 > 7 z - ---- + 22 a z + 20 a z + 10 a z - ---- - 12 a z - 16 a z - 2 a a 6 7 5 5 7 5 6 z 2 6 4 6 6 6 z > 14 a z - 6 a z - 5 z + -- - 8 a z - 8 a z - 6 a z + -- + 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 6 8 > 2 a z + 2 a z + 2 a z + a z + z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-3 3 1 1 1 1 1 1 1 q + - + q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 11 6 11 5 9 4 7 4 9 3 5 3 q t q t q t q t q t q t q t 1 4 1 2 1 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + --- + - + q t + q t + q t + q t + 7 2 5 2 3 2 5 q t q q t q t q t q t 5 4 7 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n414

L11n413 L11n415