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L11a477 L11a479

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X16,8,5,7 X18,9,19,10 X22,15,17,16 X14,19,15,20 X20,13,21,14 X12,21,13,22 X8,17,9,18 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-9 - 2q-8 + 6q-7 - 10q-6 + 15q-5 - 17q-4 + 19q-3 - 16q-2 + 13q-1 - 8 + 4q - q2

A2 (sl(3)) Invariant:

q-28 + q-26 + q-24 + 5q-22 + q-20 + 2q-18 + 6q-16 + 5q-12 + q-10 + q-8 + 3q-6 - 3q-4 + 4q-2 - 1 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 + a2 + 2a2z2 + 2a2z4 + a2z6 + a4z-2 + 3a4 + 3a4z2 + 2a4z4 + a4z6 - 2a6z-2 - 6a6 - 5a6z2 - 2a6z4 + a8z-2 + 2a8 + a8z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + 2z2 - 6z4 + 4z6 - az + 4az3 - 11az5 + 7az7 - 8a2z6 + 7a2z8 - 2a3z + 15a3z3 - 19a3z5 + 4a3z7 + 4a3z9 - a4z-2 + 7a4 - 21a4z2 + 36a4z4 - 32a4z6 + 12a4z8 + a4z10 + 2a5z-1 - 9a5z + 12a5z3 - 2a5z5 - 9a5z7 + 7a5z9 - 2a6z-2 + 11a6 - 27a6z2 + 37a6z4 - 27a6z6 + 8a6z8 + a6z10 + 2a7z-1 - 7a7z + 4a7z3 - 4a7z7 + 3a7z9 - a8z-2 + 3a8 - 3a8z2 + 3a8z4 - 6a8z6 + 3a8z8 + a9z + 2a9z3 - 5a9z5 + 2a9z7 - 2a10 + 5a10z2 - 4a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     3   j = 1                   5 1   j = -1                 8 3     j = -3               10 7       j = -5             9 6         j = -7           8 10           j = -9         7 9             j = -11       3 8               j = -13     3 7                 j = -15   1 5                   j = -17   1                     j = -19 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, Alternating, 478]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 478]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[16, 8, 5, 7], X[18, 9, 19, 10], > X[22, 15, 17, 16], X[14, 19, 15, 20], X[20, 13, 21, 14], X[12, 21, 13, 22], > X[8, 17, 9, 18], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {9, -4, 6, -7, 8, -5}, > {10, -1, 3, -9, 4, -2, 11, -8, 7, -6, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 6 10 15 17 19 16 13 2 -8 + q - -- + -- - -- + -- - -- + -- - -- + -- + 4 q - q 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 -24 5 -20 2 6 5 -10 -8 3 3 -1 + q + q + q + --- + q + --- + --- + --- + q + q + -- - -- + 22 18 16 12 6 4 q q q q q q 4 2 4 6 > -- - q + 2 q - q 2 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 478]][a, z]
Out[8]=	4 6 8 2 4 6 8 a 2 a a 2 2 2 4 2 6 2 a + 3 a - 6 a + 2 a + -- - ---- + -- - z + 2 a z + 3 a z - 5 a z + 2 2 2 z z z 8 2 4 2 4 4 4 6 4 2 6 4 6 > a z - z + 2 a z + 2 a z - 2 a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 478]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 10 a 2 a a 2 a 2 a 3 7 a + 11 a + 3 a - 2 a - -- - ---- - -- + ---- + ---- - a z - 2 a z - 2 2 2 z z z z z 5 7 9 2 4 2 6 2 8 2 10 2 > 9 a z - 7 a z + a z + 2 z - 21 a z - 27 a z - 3 a z + 5 a z - 3 z 3 3 3 5 3 7 3 9 3 4 4 4 > -- + 4 a z + 15 a z + 12 a z + 4 a z + 2 a z - 6 z + 36 a z + a 5 6 4 8 4 10 4 z 5 3 5 5 5 > 37 a z + 3 a z - 4 a z + -- - 11 a z - 19 a z - 2 a z - a 9 5 6 2 6 4 6 6 6 8 6 10 6 > 5 a z + 4 z - 8 a z - 32 a z - 27 a z - 6 a z + a z + 7 3 7 5 7 7 7 9 7 2 8 4 8 > 7 a z + 4 a z - 9 a z - 4 a z + 2 a z + 7 a z + 12 a z + 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 8 a z + 3 a z + 4 a z + 7 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 8 1 1 1 5 3 7 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 8 7 9 8 10 9 6 10 3 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a478

L11a477 L11a479