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

K11a341 K11a343

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X14,4,15,3 X16,6,17,5 X22,8,1,7 X20,10,21,9 X18,12,19,11 X2,14,3,13 X4,16,5,15 X12,18,13,17 X10,20,11,19 X8,22,9,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 14 16 22 20 18 2 4 12 10 8

Alexander Polynomial:

4t-2 - 7t-1 + 7 - 7t + 4t2

Conway Polynomial:

1 + 9z2 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{29, 4}

Jones Polynomial:

q2 - q3 + 2q4 - 3q5 + 4q6 - 4q7 + 4q8 - 3q9 + 3q10 - 2q11 + q12 - q13

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

{...}

A2 (sl(3)) Invariant:

q6 + q10 + q16 + q26 + q28 + q30 + q32 - q34 - q36 - q38 - q40

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 4a-15z + 10a-15z3 - 6a-15z5 + a-15z7 - a-14z2 + 6a-14z4 - 5a-14z6 + a-14z8 + a-13z - 4a-13z3 + 7a-13z5 - 5a-13z7 + a-13z9 - 2a-12 + 16a-12z2 - 29a-12z4 + 20a-12z6 - 7a-12z8 + a-12z10 + 5a-11z - 24a-11z3 + 26a-11z5 - 12a-11z7 + 2a-11z9 - 2a-10 + 13a-10z2 - 26a-10z4 + 20a-10z6 - 7a-10z8 + a-10z10 - 4a-9z3 + 9a-9z5 - 5a-9z7 + a-9z9 + 6a-8z4 - 4a-8z6 + a-8z8 + 4a-7z3 - 3a-7z5 + a-7z7 + a-6z2 - 2a-6z4 + a-6z6 - 2a-5z3 + a-5z5 + a-4 - 3a-4z2 + a-4z4

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

{9, 29}

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 11342. 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 = 27                       1 j = 25                         j = 23                   2 1   j = 21                 1       j = 19               2 2       j = 17             2 1         j = 15           2 2           j = 13         2 2             j = 11       1 2               j = 9     1 2                 j = 7     1                   j = 5 1 1                     j = 3 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, 342]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 342]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 4, 15, 3], X[16, 6, 17, 5], X[22, 8, 1, 7], > X[20, 10, 21, 9], X[18, 12, 19, 11], X[2, 14, 3, 13], X[4, 16, 5, 15], > X[12, 18, 13, 17], X[10, 20, 11, 19], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 342]]
Out[4]=	GaussCode[1, -7, 2, -8, 3, -1, 4, -11, 5, -10, 6, -9, 7, -2, 8, -3, 9, -6, 10, > -5, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 342]]
Out[5]=	DTCode[6, 14, 16, 22, 20, 18, 2, 4, 12, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 342]][t]
Out[6]=	4 7 2 7 + -- - - - 7 t + 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 342]][z]
Out[7]=	2 4 1 + 9 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 342]}
In[9]:=	{KnotDet[Knot[11, Alternating, 342]], KnotSignature[Knot[11, Alternating, 342]]}
Out[9]=	{29, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 342]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 12 13 q - q + 2 q - 3 q + 4 q - 4 q + 4 q - 3 q + 3 q - 2 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 342]}
In[12]:=	A2Invariant[Knot[11, Alternating, 342]][q]
Out[12]=	6 10 16 26 28 30 32 34 36 38 40 q + q + q + q + q + q + q - q - q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 342]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 4 -2 2 -4 z 3 z 2 z 2 z 3 z z z z z --- + --- + a - --- + ---- + ---- + ---- + ---- + --- + -- + -- + -- 12 10 12 10 8 6 4 10 8 6 4 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 342]][a, z]
Out[14]=	2 2 2 2 2 3 -2 2 -4 4 z z 5 z z 16 z 13 z z 3 z 10 z --- - --- + a - --- + --- + --- - --- + ----- + ----- + -- - ---- + ----- - 12 10 15 13 11 14 12 10 6 4 15 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 4 z 24 z 4 z 4 z 2 z 6 z 29 z 26 z 6 z 2 z > ---- - ----- - ---- + ---- - ---- + ---- - ----- - ----- + ---- - ---- + 13 11 9 7 5 14 12 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 5 6 6 6 6 z 6 z 7 z 26 z 9 z 3 z z 5 z 20 z 20 z 4 z > -- - ---- + ---- + ----- + ---- - ---- + -- - ---- + ----- + ----- - ---- + 4 15 13 11 9 7 5 14 12 10 8 a a a a a a a a a a a 6 7 7 7 7 7 8 8 8 8 9 9 z z 5 z 12 z 5 z z z 7 z 7 z z z 2 z > -- + --- - ---- - ----- - ---- + -- + --- - ---- - ---- + -- + --- + ---- + 6 15 13 11 9 7 14 12 10 8 13 11 a a a a a a a a a a a a 9 10 10 z z z > -- + --- + --- 9 12 10 a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 342]], Vassiliev[3][Knot[11, Alternating, 342]]}
Out[15]=	{9, 29}
In[16]:=	Kh[Knot[11, Alternating, 342]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 13 4 q + q + q t + q t + q t + 2 q t + q t + 2 q t + 2 q t + 13 5 15 5 15 6 17 6 17 7 19 7 19 8 > 2 q t + 2 q t + 2 q t + 2 q t + q t + 2 q t + 2 q t + 21 8 23 9 23 10 27 11 > q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a342

K11a341 K11a343