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K11a342 K11a344

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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 X4,14,5,13 X2,16,3,15 X12,18,13,17 X10,20,11,19 X8,22,9,21

Gauss Code:

{1, -8, 2, -7, 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 4 2 12 10 8

Alexander Polynomial:

8t-1 - 15 + 8t

Conway Polynomial:

1 + 8z2

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{31, 2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

q2 - q4 + q8 + q12 + q14 + q22 + q24 + q28 - q32 - q36 - q38

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

8a-13z - 24a-13z3 + 22a-13z5 - 8a-13z7 + a-13z9 - a-12 + 5a-12z2 - 14a-12z4 + 16a-12z6 - 7a-12z8 + a-12z10 + 8a-11z - 34a-11z3 + 43a-11z5 - 20a-11z7 + 3a-11z9 - 5a-10z2 + 10a-10z4 - 4a-10z8 + a-10z10 + 9a-9z5 - 9a-9z7 + 2a-9z9 + a-8 - 6a-8z2 + 15a-8z4 - 13a-8z6 + 3a-8z8 + 4a-7z3 - 9a-7z5 + 3a-7z7 - 6a-6z4 + 3a-6z6 - 4a-5z3 + 3a-5z5 + a-4 - 3a-4z2 + 3a-4z4 + 2a-3z3 + a-2z2

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

{8, 24}

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 11343. 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 = 25                       1 j = 23                         j = 21                   2 1   j = 19                 1       j = 17               2 2       j = 15             2 1         j = 13           2 2           j = 11         2 2             j = 9       1 2               j = 7     2 2                 j = 5     1                   j = 3 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, 343]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 343]]
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[4, 14, 5, 13], X[2, 16, 3, 15], > X[12, 18, 13, 17], X[10, 20, 11, 19], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 343]]
Out[4]=	GaussCode[1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -9, 7, -2, 8, -3, 9, -6, 10, > -5, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 343]]
Out[5]=	DTCode[6, 14, 16, 22, 20, 18, 4, 2, 12, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 343]][t]
Out[6]=	8 -15 + - + 8 t t
In[7]:=	Conway[Knot[11, Alternating, 343]][z]
Out[7]=	2 1 + 8 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 343]}
In[9]:=	{KnotDet[Knot[11, Alternating, 343]], KnotSignature[Knot[11, Alternating, 343]]}
Out[9]=	{31, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 343]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 12 q - 2 q + 3 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, 343]}
In[12]:=	A2Invariant[Knot[11, Alternating, 343]][q]
Out[12]=	2 4 8 12 14 22 24 28 32 36 38 q - q + q + q + q + q + q + q - q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 343]][a, z]
Out[13]=	2 2 2 2 2 -12 -8 -4 z 2 z 2 z 2 z z -a + a + a + --- + ---- + ---- + ---- + -- 10 8 6 4 2 a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 343]][a, z]
Out[14]=	2 2 2 2 2 3 -12 -8 -4 8 z 8 z 5 z 5 z 6 z 3 z z 24 z -a + a + a + --- + --- + ---- - ---- - ---- - ---- + -- - ----- - 13 11 12 10 8 4 2 13 a a a a a a a a 3 3 3 3 4 4 4 4 4 5 34 z 4 z 4 z 2 z 14 z 10 z 15 z 6 z 3 z 22 z > ----- + ---- - ---- + ---- - ----- + ----- + ----- - ---- + ---- + ----- + 11 7 5 3 12 10 8 6 4 13 a a a a a a a a a a 5 5 5 5 6 6 6 7 7 7 43 z 9 z 9 z 3 z 16 z 13 z 3 z 8 z 20 z 9 z > ----- + ---- - ---- + ---- + ----- - ----- + ---- - ---- - ----- - ---- + 11 9 7 5 12 8 6 13 11 9 a a a a a a a a a a 7 8 8 8 9 9 9 10 10 3 z 7 z 4 z 3 z z 3 z 2 z z z > ---- - ---- - ---- + ---- + --- + ---- + ---- + --- + --- 7 12 10 8 13 11 9 12 10 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 343]], Vassiliev[3][Knot[11, Alternating, 343]]}
Out[15]=	{8, 24}
In[16]:=	Kh[Knot[11, Alternating, 343]][q, t]
Out[16]=	3 3 5 2 7 2 7 3 9 3 9 4 11 4 q + q + 2 q t + q t + 2 q t + 2 q t + q t + 2 q t + 2 q t + 11 5 13 5 13 6 15 6 15 7 17 7 17 8 > 2 q t + 2 q t + 2 q t + 2 q t + q t + 2 q t + 2 q t + 19 8 21 9 21 10 25 11 > q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a343

K11a342 K11a344