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

K11a5 K11a7

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X18,12,19,11 X20,13,21,14 X6,16,7,15 X14,18,15,17 X22,20,1,19 X12,21,13,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 16 2 18 20 6 14 22 12

Alexander Polynomial:

2t-3 - 13t-2 + 32t-1 - 41 + 32t - 13t2 + 2t3

Conway Polynomial:

1 - 2z2 - z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a132, K11a352, ...}

Determinant and Signature:

{135, 2}

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 12 + 18q - 21q2 + 22q3 - 20q4 + 15q5 - 10q6 + 5q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 - 2q-2 - 1 + 4q2 - 3q4 + 4q6 - q8 - q10 + q12 - 5q14 + 4q16 - q18 - q20 + 3q22 - q24

HOMFLY-PT Polynomial:

a-6 - a-6z4 - 2a-4 - a-4z2 + a-4z4 + a-4z6 + 2a-2 + a-2z2 + a-2z4 + a-2z6 - 1 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

a-9z5 - 5a-8z4 + 5a-8z6 + 5a-7z3 - 15a-7z5 + 10a-7z7 - a-6 + 4a-6z2 - 4a-6z4 - 10a-6z6 + 10a-6z8 - 4a-5z + 24a-5z3 - 36a-5z5 + 11a-5z7 + 5a-5z9 - 2a-4 + 4a-4z2 + 13a-4z4 - 34a-4z6 + 18a-4z8 + a-4z10 - 8a-3z + 32a-3z3 - 33a-3z5 + 2a-3z7 + 8a-3z9 - 2a-2 + a-2z2 + 14a-2z4 - 26a-2z6 + 12a-2z8 + a-2z10 - 6a-1z + 20a-1z3 - 21a-1z5 + 4a-1z7 + 3a-1z9 - 1 + 4z2 - z4 - 6z6 + 4z8 - 2az + 7az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + 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 116. 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                     4   j = 13                   6 1   j = 11                 9 4     j = 9               11 6       j = 7             11 9         j = 5           10 11           j = 3         8 11             j = 1       5 11               j = -1     2 7                 j = -3   1 5                   j = -5   2                     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, 6]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 6]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[16, 8, 17, 7], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[20, 13, 21, 14], X[6, 16, 7, 15], > X[14, 18, 15, 17], X[22, 20, 1, 19], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 6]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -11, 7, -9, 8, -4, 9, -6, 10, > -7, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 6]]
Out[5]=	DTCode[4, 8, 10, 16, 2, 18, 20, 6, 14, 22, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 6]][t]
Out[6]=	2 13 32 2 3 -41 + -- - -- + -- + 32 t - 13 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 6]][z]
Out[7]=	2 4 6 1 - 2 z - z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 6], Knot[11, Alternating, 132], > Knot[11, Alternating, 352]}
In[9]:=	{KnotDet[Knot[11, Alternating, 6]], KnotSignature[Knot[11, Alternating, 6]]}
Out[9]=	{135, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 6]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 8 -12 + q - -- + - + 18 q - 21 q + 22 q - 20 q + 15 q - 10 q + 5 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 6]}
In[12]:=	A2Invariant[Knot[11, Alternating, 6]][q]
Out[12]=	-10 -6 3 2 2 4 6 8 10 12 14 -1 + q - q + -- - -- + 4 q - 3 q + 4 q - q - q + q - 5 q + 4 2 q q 16 18 20 22 24 > 4 q - q - q + 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 6]][a, z]
Out[13]=	2 2 4 4 4 6 6 -6 2 2 2 2 z z 2 2 4 z z z z z -1 + a - -- + -- + a - 3 z - -- + -- + a z - 2 z - -- + -- + -- + -- + -- 4 2 4 2 6 4 2 4 2 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 6]][a, z]
Out[14]=	2 2 2 -6 2 2 2 4 z 8 z 6 z 2 4 z 4 z z -1 - a - -- - -- - a - --- - --- - --- - 2 a z + 4 z + ---- + ---- + -- + 4 2 5 3 a 6 4 2 a a a a a a a 3 3 3 3 4 4 2 2 5 z 24 z 32 z 20 z 3 4 5 z 4 z > 3 a z + ---- + ----- + ----- + ----- + 7 a z - z - ---- - ---- + 7 5 3 a 8 6 a a a a a 4 4 5 5 5 5 5 13 z 14 z 2 4 z 15 z 36 z 33 z 21 z 5 > ----- + ----- - 3 a z + -- - ----- - ----- - ----- - ----- - 8 a z - 4 2 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 7 6 5 z 10 z 34 z 26 z 2 6 10 z 11 z 2 z 4 z > 6 z + ---- - ----- - ----- - ----- + a z + ----- + ----- + ---- + ---- + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 10 z 18 z 12 z 5 z 8 z 3 z z z > 3 a z + 4 z + ----- + ----- + ----- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 6]], Vassiliev[3][Knot[11, Alternating, 6]]}
Out[15]=	{-2, -2}
In[16]:=	Kh[Knot[11, Alternating, 6]][q, t]
Out[16]=	3 1 2 1 5 2 7 5 q 3 11 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 > 10 q t + 11 q t + 11 q t + 9 q t + 11 q t + 6 q t + 9 q t + 11 5 13 5 13 6 15 6 17 7 > 4 q t + 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a6

K11a5 K11a7