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

K11a355 K11a357

Knotscape

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DrawMorseLink

PD Presentation:

X8291 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X6,16,7,15 X4,18,5,17 X12,20,13,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

8 14 16 18 20 22 2 6 4 12 10

Alexander Polynomial:

4t-3 - 10t-2 + 16t-1 - 19 + 16t - 10t2 + 4t3

Conway Polynomial:

1 + 12z2 + 14z4 + 4z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{79, 6}

Jones Polynomial:

q3 - 2q4 + 5q5 - 8q6 + 11q7 - 12q8 + 13q9 - 11q10 + 8q11 - 5q12 + 2q13 - q14

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

{...}

A2 (sl(3)) Invariant:

q10 - q12 + 2q14 - q16 + 2q20 - q22 + 4q24 + q28 + q30 - 2q32 + q34 - 2q36 - q38 - q42

HOMFLY-PT Polynomial:

- 2a-12 - 3a-12z2 - a-12z4 + 2a-10z2 + 3a-10z4 + a-10z6 + 3a-8 + 9a-8z2 + 8a-8z4 + 2a-8z6 + 4a-6z2 + 4a-6z4 + a-6z6

Kauffman Polynomial:

2a-17z - 3a-17z3 + a-17z5 + a-16z2 - 4a-16z4 + 2a-16z6 - 3a-15z + 5a-15z3 - 6a-15z5 + 3a-15z7 - 2a-14z2 + 5a-14z4 - 5a-14z6 + 3a-14z8 - a-13z + 4a-13z3 - a-13z5 - a-13z7 + 2a-13z9 - 2a-12 + 13a-12z2 - 13a-12z4 + 5a-12z6 + a-12z10 + 3a-11z - 10a-11z3 + 13a-11z5 - 9a-11z7 + 4a-11z9 - 4a-10z4 + a-10z6 + a-10z10 - a-9z - 2a-9z3 + a-9z5 - 3a-9z7 + 2a-9z9 + 3a-8 - 12a-8z2 + 14a-8z4 - 10a-8z6 + 3a-8z8 + 4a-7z3 - 6a-7z5 + 2a-7z7 + 4a-6z2 - 4a-6z4 + a-6z6

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

{12, 40}

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=6 is the signature of 11356. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 29                       1 j = 27                     1   j = 25                   4 1   j = 23                 4 1     j = 21               7 4       j = 19             6 4         j = 17           6 7           j = 15         5 6             j = 13       3 6               j = 11     2 5                 j = 9     3                   j = 7 1 2                     j = 5 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, 356]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 356]]
Out[3]=	PD[X[8, 2, 9, 1], X[14, 4, 15, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[20, 10, 21, 9], X[22, 12, 1, 11], X[2, 14, 3, 13], X[6, 16, 7, 15], > X[4, 18, 5, 17], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 356]]
Out[4]=	GaussCode[1, -7, 2, -9, 3, -8, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 356]]
Out[5]=	DTCode[8, 14, 16, 18, 20, 22, 2, 6, 4, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 356]][t]
Out[6]=	4 10 16 2 3 -19 + -- - -- + -- + 16 t - 10 t + 4 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 356]][z]
Out[7]=	2 4 6 1 + 12 z + 14 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 356]}
In[9]:=	{KnotDet[Knot[11, Alternating, 356]], KnotSignature[Knot[11, Alternating, 356]]}
Out[9]=	{79, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 356]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 2 q + 5 q - 8 q + 11 q - 12 q + 13 q - 11 q + 8 q - 5 q + 13 14 > 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 356]}
In[12]:=	A2Invariant[Knot[11, Alternating, 356]][q]
Out[12]=	10 12 14 16 20 22 24 28 30 32 34 q - q + 2 q - q + 2 q - q + 4 q + q + q - 2 q + q - 36 38 42 > 2 q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 356]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 6 6 -2 3 3 z 2 z 9 z 4 z z 3 z 8 z 4 z z 2 z --- + -- - ---- + ---- + ---- + ---- - --- + ---- + ---- + ---- + --- + ---- + 12 8 12 10 8 6 12 10 8 6 10 8 a a a a a a a a a a a a 6 z > -- 6 a
In[14]:=	Kauffman[Knot[11, Alternating, 356]][a, z]
Out[14]=	2 2 2 2 2 -2 3 2 z 3 z z 3 z z z 2 z 13 z 12 z 4 z --- + -- + --- - --- - --- + --- - -- + --- - ---- + ----- - ----- + ---- - 12 8 17 15 13 11 9 16 14 12 8 6 a a a a a a a a a a a a 3 3 3 3 3 3 4 4 4 4 3 z 5 z 4 z 10 z 2 z 4 z 4 z 5 z 13 z 4 z > ---- + ---- + ---- - ----- - ---- + ---- - ---- + ---- - ----- - ---- + 17 15 13 11 9 7 16 14 12 10 a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 6 14 z 4 z z 6 z z 13 z z 6 z 2 z 5 z 5 z > ----- - ---- + --- - ---- - --- + ----- + -- - ---- + ---- - ---- + ---- + 8 6 17 15 13 11 9 7 16 14 12 a a a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 9 z 10 z z 3 z z 9 z 3 z 2 z 3 z 3 z 2 z > --- - ----- + -- + ---- - --- - ---- - ---- + ---- + ---- + ---- + ---- + 10 8 6 15 13 11 9 7 14 8 13 a a a a a a a a a a a 9 9 10 10 4 z 2 z z z > ---- + ---- + --- + --- 11 9 12 10 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 356]], Vassiliev[3][Knot[11, Alternating, 356]]}
Out[15]=	{12, 40}
In[16]:=	Kh[Knot[11, Alternating, 356]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 2 q t + 3 q t + 2 q t + 5 q t + 3 q t + 6 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 5 q t + 6 q t + 6 q t + 7 q t + 6 q t + 4 q t + 21 7 21 8 23 8 23 9 25 9 25 10 27 10 > 7 q t + 4 q t + 4 q t + q t + 4 q t + q t + q t + 29 11 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a356

K11a355 K11a357