© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a276 K11a278

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,4,11,3 X16,5,17,6 X12,8,13,7 X4,10,5,9 X18,11,19,12 X20,14,21,13 X22,16,1,15 X2,17,3,18 X8,19,9,20 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 12 4 18 20 22 2 8 14

Alexander Polynomial:

- t-4 + 6t-3 - 17t-2 + 28t-1 - 31 + 28t - 17t2 + 6t3 - t4

Conway Polynomial:

1 - 2z2 - z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a99, ...}

Determinant and Signature:

{135, 2}

Jones Polynomial:

q-3 - 4q-2 + 9q-1 - 13 + 19q - 22q2 + 21q3 - 19q4 + 14q5 - 8q6 + 4q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-8 - 2q-6 + 3q-4 + 1 + 4q2 - 5q4 + 3q6 - 4q8 + q12 - 3q14 + 4q16 - q18 + q20 + q22 - q24

HOMFLY-PT Polynomial:

- 2a-6z2 - a-6z4 + 2a-4 + 7a-4z2 + 7a-4z4 + 2a-4z6 - 4a-2 - 10a-2z2 - 10a-2z4 - 5a-2z6 - a-2z8 + 3 + 3z2 + 3z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 4a-8z6 - a-7z + 3a-7z3 - 10a-7z5 + 7a-7z7 + 3a-6z2 - 2a-6z4 - 9a-6z6 + 8a-6z8 - 5a-5z + 6a-5z3 - 8a-5z7 + 7a-5z9 + 2a-4 - 11a-4z2 + 31a-4z4 - 28a-4z6 + 7a-4z8 + 3a-4z10 - 5a-3z - 6a-3z3 + 38a-3z5 - 40a-3z7 + 15a-3z9 + 4a-2 - 21a-2z2 + 49a-2z4 - 39a-2z6 + 7a-2z8 + 3a-2z10 - a-1z - 6a-1z3 + 18a-1z5 - 21a-1z7 + 8a-1z9 + 3 - 9z2 + 20z4 - 23z6 + 8z8 + 2az3 - 9az5 + 4az7 - 2a2z4 + a2z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-2, -2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 11277. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   5 1   j = 11                 9 3     j = 9               10 5       j = 7             11 9         j = 5           11 10           j = 3         8 11             j = 1       6 12               j = -1     3 7                 j = -3   1 6                   j = -5   3                     j = -7 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 277]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 277]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[16, 5, 17, 6], X[12, 8, 13, 7], > X[4, 10, 5, 9], X[18, 11, 19, 12], X[20, 14, 21, 13], X[22, 16, 1, 15], > X[2, 17, 3, 18], X[8, 19, 9, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 277]]
Out[4]=	GaussCode[1, -9, 2, -5, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -6, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 277]]
Out[5]=	DTCode[6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 277]][t]
Out[6]=	-4 6 17 28 2 3 4 -31 - t + -- - -- + -- + 28 t - 17 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 277]][z]
Out[7]=	2 4 6 8 1 - 2 z - z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 99], Knot[11, Alternating, 277]}
In[9]:=	{KnotDet[Knot[11, Alternating, 277]], KnotSignature[Knot[11, Alternating, 277]]}
Out[9]=	{135, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 277]][q]
Out[10]=	-3 4 9 2 3 4 5 6 7 8 -13 + q - -- + - + 19 q - 22 q + 21 q - 19 q + 14 q - 8 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 277]}
In[12]:=	A2Invariant[Knot[11, Alternating, 277]][q]
Out[12]=	-8 2 3 2 4 6 8 12 14 16 18 1 + q - -- + -- + 4 q - 5 q + 3 q - 4 q + q - 3 q + 4 q - q + 6 4 q q 20 22 24 > q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 277]][a, z]
Out[13]=	2 2 2 4 4 4 2 4 2 2 z 7 z 10 z 4 z 7 z 10 z 6 3 + -- - -- + 3 z - ---- + ---- - ----- + 3 z - -- + ---- - ----- + z + 4 2 6 4 2 6 4 2 a a a a a a a a 6 6 8 2 z 5 z z > ---- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 277]][a, z]
Out[14]=	2 2 2 2 3 2 4 z 5 z 5 z z 2 2 z 3 z 11 z 21 z z 3 + -- + -- - -- - --- - --- - - - 9 z + ---- + ---- - ----- - ----- - -- + 4 2 7 5 3 a 8 6 4 2 9 a a a a a a a a a a 3 3 3 3 4 4 4 4 3 z 6 z 6 z 6 z 3 4 6 z 2 z 31 z 49 z > ---- + ---- - ---- - ---- + 2 a z + 20 z - ---- - ---- + ----- + ----- - 7 5 3 a 8 6 4 2 a a a a a a a 5 5 5 5 6 6 2 4 z 10 z 38 z 18 z 5 6 4 z 9 z > 2 a z + -- - ----- + ----- + ----- - 9 a z - 23 z + ---- - ---- - 9 7 3 a 8 6 a a a a a 6 6 7 7 7 7 28 z 39 z 2 6 7 z 8 z 40 z 21 z 7 8 > ----- - ----- + a z + ---- - ---- - ----- - ----- + 4 a z + 8 z + 4 2 7 5 3 a a a a a a 8 8 8 9 9 9 10 10 8 z 7 z 7 z 7 z 15 z 8 z 3 z 3 z > ---- + ---- + ---- + ---- + ----- + ---- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 277]], Vassiliev[3][Knot[11, Alternating, 277]]}
Out[15]=	{-2, -2}
In[16]:=	Kh[Knot[11, Alternating, 277]][q, t]
Out[16]=	3 1 3 1 6 3 7 6 q 3 12 q + 8 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 11 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 11 q t + 10 q t + 11 q t + 9 q t + 10 q t + 5 q t + 9 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a277

K11a276 K11a278