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

K11a345 K11a347

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 16 12 18 20 4 22 10 2 8 14

Alexander Polynomial:

t-4 - 5t-3 + 12t-2 - 18t-1 + 21 - 18t + 12t2 - 5t3 + t4

Conway Polynomial:

1 + z2 + 2z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a106, K11a194, ...}

Determinant and Signature:

{93, 4}

Jones Polynomial:

q-2 - 3q-1 + 6 - 9q + 13q2 - 14q3 + 14q4 - 13q5 + 10q6 - 6q7 + 3q8 - q9

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

{...}

A2 (sl(3)) Invariant:

q-6 + 1 - 2q2 + 2q4 + 2q10 - 3q12 + 3q14 - q16 + q18 + q20 - 2q22 + q24 - q26

HOMFLY-PT Polynomial:

- 2a-6 - 5a-6z2 - 4a-6z4 - a-6z6 + 5a-4 + 15a-4z2 + 14a-4z4 + 6a-4z6 + a-4z8 - 4a-2 - 12a-2z2 - 9a-2z4 - 2a-2z6 + 2 + 3z2 + z4

Kauffman Polynomial:

a-11z3 + 3a-10z4 - 3a-9z3 + 6a-9z5 + 4a-8z2 - 14a-8z4 + 10a-8z6 - 4a-7z + 17a-7z3 - 29a-7z5 + 13a-7z7 + 2a-6 - 7a-6z2 + 19a-6z4 - 31a-6z6 + 12a-6z8 - 7a-5z + 27a-5z3 - 17a-5z5 - 12a-5z7 + 7a-5z9 + 5a-4 - 30a-4z2 + 68a-4z4 - 53a-4z6 + 8a-4z8 + 2a-4z10 - 3a-3z - 6a-3z3 + 42a-3z5 - 40a-3z7 + 10a-3z9 + 4a-2 - 26a-2z2 + 41a-2z4 - 17a-2z6 - 3a-2z8 + 2a-2z10 - 12a-1z3 + 24a-1z5 - 15a-1z7 + 3a-1z9 + 2 - 7z2 + 9z4 - 5z6 + z8

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

{1, 3}

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 11346. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                       1 j = 17                     2   j = 15                   4 1   j = 13                 6 2     j = 11               7 4       j = 9             7 6         j = 7           7 7           j = 5         6 7             j = 3       4 8               j = 1     2 5                 j = -1   1 4                   j = -3   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, 346]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 346]]
Out[3]=	PD[X[6, 2, 7, 1], X[16, 3, 17, 4], X[12, 6, 13, 5], X[18, 8, 19, 7], > X[20, 10, 21, 9], X[4, 12, 5, 11], X[22, 13, 1, 14], X[10, 16, 11, 15], > X[2, 17, 3, 18], X[8, 20, 9, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 346]]
Out[4]=	GaussCode[1, -9, 2, -6, 3, -1, 4, -10, 5, -8, 6, -3, 7, -11, 8, -2, 9, -4, 10, > -5, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 346]]
Out[5]=	DTCode[6, 16, 12, 18, 20, 4, 22, 10, 2, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 346]][t]
Out[6]=	-4 5 12 18 2 3 4 21 + t - -- + -- - -- - 18 t + 12 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 346]][z]
Out[7]=	2 4 6 8 1 + z + 2 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 106], Knot[11, Alternating, 194], > Knot[11, Alternating, 346]}
In[9]:=	{KnotDet[Knot[11, Alternating, 346]], KnotSignature[Knot[11, Alternating, 346]]}
Out[9]=	{93, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 346]][q]
Out[10]=	-2 3 2 3 4 5 6 7 8 9 6 + q - - - 9 q + 13 q - 14 q + 14 q - 13 q + 10 q - 6 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 346]}
In[12]:=	A2Invariant[Knot[11, Alternating, 346]][q]
Out[12]=	-6 2 4 10 12 14 16 18 20 22 24 1 + q - 2 q + 2 q + 2 q - 3 q + 3 q - q + q + q - 2 q + q - 26 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 346]][a, z]
Out[13]=	2 2 2 4 4 4 2 5 4 2 5 z 15 z 12 z 4 4 z 14 z 9 z 2 - -- + -- - -- + 3 z - ---- + ----- - ----- + z - ---- + ----- - ---- - 6 4 2 6 4 2 6 4 2 a a a a a a a a a 6 6 6 8 z 6 z 2 z z > -- + ---- - ---- + -- 6 4 2 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 346]][a, z]
Out[14]=	2 2 2 2 3 2 5 4 4 z 7 z 3 z 2 4 z 7 z 30 z 26 z z 2 + -- + -- + -- - --- - --- - --- - 7 z + ---- - ---- - ----- - ----- + --- - 6 4 2 7 5 3 8 6 4 2 11 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 3 z 17 z 27 z 6 z 12 z 4 3 z 14 z 19 z 68 z > ---- + ----- + ----- - ---- - ----- + 9 z + ---- - ----- + ----- + ----- + 9 7 5 3 a 10 8 6 4 a a a a a a a a 4 5 5 5 5 5 6 6 41 z 6 z 29 z 17 z 42 z 24 z 6 10 z 31 z > ----- + ---- - ----- - ----- + ----- + ----- - 5 z + ----- - ----- - 2 9 7 5 3 a 8 6 a a a a a a a 6 6 7 7 7 7 8 8 8 53 z 17 z 13 z 12 z 40 z 15 z 8 12 z 8 z 3 z > ----- - ----- + ----- - ----- - ----- - ----- + z + ----- + ---- - ---- + 4 2 7 5 3 a 6 4 2 a a a a a a a a 9 9 9 10 10 7 z 10 z 3 z 2 z 2 z > ---- + ----- + ---- + ----- + ----- 5 3 a 4 2 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 346]], Vassiliev[3][Knot[11, Alternating, 346]]}
Out[15]=	{1, 3}
In[16]:=	Kh[Knot[11, Alternating, 346]][q, t]
Out[16]=	3 3 5 1 2 1 4 2 q 5 q 4 q 5 8 q + 6 q + ----- + ----- + ---- + ---- + --- + --- + ---- + 7 q t + 5 4 3 3 3 2 2 t t q t q t q t q t t 7 7 2 9 2 9 3 11 3 11 4 13 4 > 7 q t + 7 q t + 7 q t + 6 q t + 7 q t + 4 q t + 6 q t + 13 5 15 5 15 6 17 6 19 7 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a346

K11a345 K11a347