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

K11a356 K11a358

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 X6,14,7,13 X4,16,5,15 X2,18,3,17 X12,20,13,19 X10,22,11,21

Gauss Code:

{1, -9, 2, -8, 3, -7, 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 6 4 2 12 10

Alexander Polynomial:

4t-3 - 11t-2 + 19t-1 - 23 + 19t - 11t2 + 4t3

Conway Polynomial:

1 + 11z2 + 13z4 + 4z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{91, 6}

Jones Polynomial:

q3 - 2q4 + 5q5 - 8q6 + 12q7 - 14q8 + 15q9 - 13q10 + 10q11 - 7q12 + 3q13 - q14

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

{...}

A2 (sl(3)) Invariant:

q10 - q12 + 2q14 - q16 + 3q20 - q22 + 4q24 + q30 - 3q32 + q34 - 2q36 - q38 + q40 - q42

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

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

{11, 35}

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 11357. 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                     2   j = 25                   5 1   j = 23                 5 2     j = 21               8 5       j = 19             7 5         j = 17           7 8           j = 15         5 7             j = 13       3 7               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, 357]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 357]]
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[6, 14, 7, 13], X[4, 16, 5, 15], > X[2, 18, 3, 17], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 357]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -7, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 357]]
Out[5]=	DTCode[8, 14, 16, 18, 20, 22, 6, 4, 2, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 357]][t]
Out[6]=	4 11 19 2 3 -23 + -- - -- + -- + 19 t - 11 t + 4 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 357]][z]
Out[7]=	2 4 6 1 + 11 z + 13 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 357]}
In[9]:=	{KnotDet[Knot[11, Alternating, 357]], KnotSignature[Knot[11, Alternating, 357]]}
Out[9]=	{91, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 357]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 2 q + 5 q - 8 q + 12 q - 14 q + 15 q - 13 q + 10 q - 7 q + 13 14 > 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 357]}
In[12]:=	A2Invariant[Knot[11, Alternating, 357]][q]
Out[12]=	10 12 14 16 20 22 24 30 32 34 36 q - q + 2 q - q + 3 q - q + 4 q + q - 3 q + q - 2 q - 38 40 42 > q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 357]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 6 -12 2 4 2 z z 10 z 4 z z 2 z 8 z 4 z z -a - --- + -- - ---- - --- + ----- + ---- - --- + ---- + ---- + ---- + --- + 10 8 12 10 8 6 12 10 8 6 10 a a a a a a a a a a a 6 6 2 z z > ---- + -- 8 6 a a
In[14]:=	Kauffman[Knot[11, Alternating, 357]][a, z]
Out[14]=	2 2 2 2 2 -12 2 4 z 3 z 2 z 2 z z 2 z 6 z 9 z 14 z -a + --- + -- + --- - --- + --- - --- + --- - ---- + ---- - ---- - ----- + 10 8 17 15 11 9 16 14 12 10 8 a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 4 z 2 z 5 z 3 z 6 z 2 z 4 z 5 z 5 z z > ---- - ---- + ---- + ---- - ---- + ---- + ---- - ---- + ---- + --- + 6 17 15 13 11 9 7 16 14 12 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 10 z 15 z 4 z z 9 z 3 z 11 z 2 z 6 z 3 z > ----- + ----- - ---- + --- - ---- - ---- + ----- - ---- - ---- + ---- - 10 8 6 17 15 13 11 9 7 16 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 8 8 z 7 z 7 z 10 z z 5 z z 10 z 2 z 2 z 5 z > ---- - ---- - ---- - ----- + -- + ---- - --- - ----- - ---- + ---- + ---- + 14 12 10 8 6 15 13 11 9 7 14 a a a a a a a a a a a 8 8 8 9 9 9 10 10 4 z 2 z 3 z 3 z 5 z 2 z z z > ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 12 10 8 13 11 9 12 10 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 357]], Vassiliev[3][Knot[11, Alternating, 357]]}
Out[15]=	{11, 35}
In[16]:=	Kh[Knot[11, Alternating, 357]][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 + 7 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 5 q t + 7 q t + 7 q t + 8 q t + 7 q t + 5 q t + 21 7 21 8 23 8 23 9 25 9 25 10 > 8 q t + 5 q t + 5 q t + 2 q t + 5 q t + q t + 27 10 29 11 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a357

K11a356 K11a358