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

K11a9 K11a11

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 16 2 18 20 22 6 14 12

Alexander Polynomial:

2t-3 - 11t-2 + 25t-1 - 31 + 25t - 11t2 + 2t3

Conway Polynomial:

1 - z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a262, ...}

Determinant and Signature:

{107, 2}

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 11 + 15q - 17q2 + 17q3 - 15q4 + 11q5 - 6q6 + 3q7 - q8

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

{K11a42, ...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + 3a-4 + 5a-4z2 + 3a-4z4 + a-4z6 - 2a-2 - 2a-2z2 + a-2z4 + a-2z6 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 3a-8z6 - 2a-7z + 7a-7z3 - 10a-7z5 + 5a-7z7 + a-6 - 5a-6z2 + 14a-6z4 - 13a-6z6 + 6a-6z8 - 4a-5z + 11a-5z3 - 4a-5z5 - 3a-5z7 + 4a-5z9 + 3a-4 - 18a-4z2 + 37a-4z4 - 30a-4z6 + 10a-4z8 + a-4z10 - 2a-3z + 2a-3z3 + 4a-3z5 - 11a-3z7 + 7a-3z9 + 2a-2 - 11a-2z2 + 19a-2z4 - 22a-2z6 + 8a-2z8 + a-2z10 - a-1z + 6a-1z3 - 11a-1z5 + 3a-1z9 + 3z2 - z4 - 7z6 + 4z8 - az + 6az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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

{-1, 1}

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 1110. 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                     2   j = 13                   4 1   j = 11                 7 2     j = 9               8 4       j = 7             9 7         j = 5           8 8           j = 3         7 9             j = 1       5 9               j = -1     2 6                 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, 10]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 10]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[16, 7, 17, 8], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[20, 14, 21, 13], X[22, 16, 1, 15], > X[6, 17, 7, 18], X[14, 20, 15, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 10]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -2, 5, -3, 6, -11, 7, -10, 8, -4, 9, -6, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 10]]
Out[5]=	DTCode[4, 8, 10, 16, 2, 18, 20, 22, 6, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 10]][t]
Out[6]=	2 11 25 2 3 -31 + -- - -- + -- + 25 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 10]][z]
Out[7]=	2 4 6 1 - z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 10], Knot[11, Alternating, 262]}
In[9]:=	{KnotDet[Knot[11, Alternating, 10]], KnotSignature[Knot[11, Alternating, 10]]}
Out[9]=	{107, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 10]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 8 -11 + q - -- + - + 15 q - 17 q + 17 q - 15 q + 11 q - 6 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 10], Knot[11, Alternating, 42]}
In[12]:=	A2Invariant[Knot[11, Alternating, 10]][q]
Out[12]=	-10 -6 3 -2 2 4 6 8 10 12 14 q - q + -- - q + 2 q - 4 q + 2 q - 2 q + q + 2 q - 2 q + 4 q 16 18 20 22 24 > 4 q - q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 10]][a, z]
Out[13]=	2 2 2 4 4 -6 3 2 2 2 2 z 5 z 2 z 2 2 4 z 3 z -a + -- - -- + a - 3 z - ---- + ---- - ---- + a z - 2 z - -- + ---- + 4 2 6 4 2 6 4 a a a a a a a 4 6 6 z z z > -- + -- + -- 2 4 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 10]][a, z]
Out[14]=	2 2 2 -6 3 2 2 2 z 4 z 2 z z 2 2 z 5 z 18 z a + -- + -- - a - --- - --- - --- - - - a z + 3 z + ---- - ---- - ----- - 4 2 7 5 3 a 8 6 4 a a a a a a a a 2 3 3 3 3 3 4 11 z 2 2 2 z 7 z 11 z 2 z 6 z 3 4 6 z > ----- + 3 a z - ---- + ---- + ----- + ---- + ---- + 6 a z - z - ---- + 2 9 7 5 3 a 8 a a a a a a 4 4 4 5 5 5 5 5 14 z 37 z 19 z 2 4 z 10 z 4 z 4 z 11 z > ----- + ----- + ----- - 3 a z + -- - ----- - ---- + ---- - ----- - 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 5 6 3 z 13 z 30 z 22 z 2 6 5 z 3 z > 8 a z - 7 z + ---- - ----- - ----- - ----- + a z + ---- - ---- - 8 6 4 2 7 5 a a a a a a 7 8 8 8 9 9 9 10 10 11 z 7 8 6 z 10 z 8 z 4 z 7 z 3 z z z > ----- + 3 a z + 4 z + ---- + ----- + ---- + ---- + ---- + ---- + --- + --- 3 6 4 2 5 3 a 4 2 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 10]], Vassiliev[3][Knot[11, Alternating, 10]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, Alternating, 10]][q, t]
Out[16]=	3 1 2 1 5 2 6 5 q 3 9 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 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 > 8 q t + 8 q t + 9 q t + 7 q t + 8 q t + 4 q t + 7 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a10

K11a9 K11a11