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

K11a99 K11a101

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X20,8,21,7 X16,10,17,9 X2,11,3,12 X22,14,1,13 X8,16,9,15 X14,18,15,17 X6,20,7,19 X18,22,19,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 20 16 2 22 8 14 6 18

Alexander Polynomial:

- 3t-3 + 15t-2 - 32t-1 + 41 - 32t + 15t2 - 3t3

Conway Polynomial:

1 + z2 - 3z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{K11a290, ...}

Determinant and Signature:

{141, 4}

Jones Polynomial:

1 - 3q + 8q2 - 13q3 + 19q4 - 22q5 + 23q6 - 21q7 + 15q8 - 10q9 + 5q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 - q2 + q4 + 3q6 - 3q8 + 5q10 - q12 - q14 + 3q16 - 4q18 + 3q20 - 4q22 - q24 + 2q26 - 3q28 + 3q30 + q32 - q34

HOMFLY-PT Polynomial:

- a-10z2 + a-8 + 5a-8z2 + 3a-8z4 - 3a-6 - 7a-6z2 - 6a-6z4 - 2a-6z6 + 2a-4 + 2a-4z2 - a-4z4 - a-4z6 + a-2 + 2a-2z2 + a-2z4

Kauffman Polynomial:

a-13z5 - 5a-12z4 + 5a-12z6 + 6a-11z3 - 15a-11z5 + 10a-11z7 + 2a-10z2 - 4a-10z4 - 9a-10z6 + 10a-10z8 - 4a-9z + 30a-9z3 - 44a-9z5 + 14a-9z7 + 5a-9z9 + a-8 - 6a-8z2 + 27a-8z4 - 44a-8z6 + 21a-8z8 + a-8z10 - 10a-7z + 37a-7z3 - 38a-7z5 + 2a-7z7 + 9a-7z9 + 3a-6 - 15a-6z2 + 36a-6z4 - 41a-6z6 + 16a-6z8 + a-6z10 - 7a-5z + 18a-5z3 - 17a-5z5 + a-5z7 + 4a-5z9 + 2a-4 - 4a-4z2 + 7a-4z4 - 10a-4z6 + 5a-4z8 - a-3z + 5a-3z3 - 7a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

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

{1, 0}

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=4 is the signature of 11100. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                       1 j = 21                     4   j = 19                   6 1   j = 17                 9 4     j = 15               12 6       j = 13             11 9         j = 11           11 12           j = 9         8 11             j = 7       5 11               j = 5     3 8                 j = 3   1 6                   j = 1   2                     j = -1 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, 100]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 100]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[16, 10, 17, 9], X[2, 11, 3, 12], X[22, 14, 1, 13], X[8, 16, 9, 15], > X[14, 18, 15, 17], X[6, 20, 7, 19], X[18, 22, 19, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 100]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -9, 8, -5, 9, -11, 10, > -4, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 100]]
Out[5]=	DTCode[4, 10, 12, 20, 16, 2, 22, 8, 14, 6, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 100]][t]
Out[6]=	3 15 32 2 3 41 - -- + -- - -- - 32 t + 15 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 100]][z]
Out[7]=	2 4 6 1 + z - 3 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 100], Knot[11, Alternating, 290]}
In[9]:=	{KnotDet[Knot[11, Alternating, 100]], KnotSignature[Knot[11, Alternating, 100]]}
Out[9]=	{141, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 100]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 3 q + 8 q - 13 q + 19 q - 22 q + 23 q - 21 q + 15 q - 10 q + 10 11 > 5 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 100]}
In[12]:=	A2Invariant[Knot[11, Alternating, 100]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 22 1 - q + q + 3 q - 3 q + 5 q - q - q + 3 q - 4 q + 3 q - 4 q - 24 26 28 30 32 34 > q + 2 q - 3 q + 3 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 100]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 4 -8 3 2 -2 z 5 z 7 z 2 z 2 z 3 z 6 z z z a - -- + -- + a - --- + ---- - ---- + ---- + ---- + ---- - ---- - -- + -- - 6 4 10 8 6 4 2 8 6 4 2 a a a a a a a a a a a 6 6 2 z z > ---- - -- 6 4 a a
In[14]:=	Kauffman[Knot[11, Alternating, 100]][a, z]
Out[14]=	2 2 2 2 -8 3 2 -2 4 z 10 z 7 z z 2 z 6 z 15 z 4 z a + -- + -- - a - --- - ---- - --- - -- + ---- - ---- - ----- - ---- + 6 4 9 7 5 3 10 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 3 z 6 z 30 z 37 z 18 z 5 z 5 z 4 z 27 z 36 z > ---- + ---- + ----- + ----- + ----- + ---- - ---- - ---- + ----- + ----- + 2 11 9 7 5 3 12 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 7 z 3 z z 15 z 44 z 38 z 17 z 7 z 5 z 9 z > ---- - ---- + --- - ----- - ----- - ----- - ----- - ---- + ---- - ---- - 4 2 13 11 9 7 5 3 12 10 a a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 44 z 41 z 10 z z 10 z 14 z 2 z z 3 z 10 z > ----- - ----- - ----- + -- + ----- + ----- + ---- + -- + ---- + ----- + 8 6 4 2 11 9 7 5 3 10 a a a a a a a a a a 8 8 8 9 9 9 10 10 21 z 16 z 5 z 5 z 9 z 4 z z z > ----- + ----- + ---- + ---- + ---- + ---- + --- + --- 8 6 4 9 7 5 8 6 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 100]], Vassiliev[3][Knot[11, Alternating, 100]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, Alternating, 100]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 6 q + 3 q + ---- + --- + -- + 8 q t + 5 q t + 11 q t + 8 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 11 q t + 11 q t + 12 q t + 11 q t + 9 q t + 12 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 6 q t + 9 q t + 4 q t + 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a100

K11a99 K11a101