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

K11n172 K11n174

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X16,6,17,5 X12,8,13,7 X20,10,21,9 X11,3,12,2 X18,13,19,14 X4,16,5,15 X22,17,1,18 X8,20,9,19 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 -10 16 12 20 -2 18 4 22 8 14

Alexander Polynomial:

t-4 - 4t-3 + 8t-2 - 7t-1 + 5 - 7t + 8t2 - 4t3 + t4

Conway Polynomial:

1 + 5z2 + 4z4 + 4z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{45, 4}

Jones Polynomial:

- q-1 + 3 - 5q + 7q2 - 7q3 + 8q4 - 6q5 + 5q6 - 3q7

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

{...}

A2 (sl(3)) Invariant:

- q-2 + 1 - q2 + q8 + 4q10 + 4q14 - q16 - q20 - 2q22 - 2q26 + q28

HOMFLY-PT Polynomial:

a-8z2 - 4a-6 - 7a-6z2 - 5a-6z4 - a-6z6 + 7a-4 + 15a-4z2 + 13a-4z4 + 6a-4z6 + a-4z8 - 2a-2 - 4a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

3a-9z - 2a-8z2 + 3a-8z4 - a-7z + 8a-7z3 - 9a-7z5 + 3a-7z7 + 4a-6 - 20a-6z2 + 30a-6z4 - 21a-6z6 + 5a-6z8 - 8a-5z + 20a-5z3 - 14a-5z5 - 2a-5z7 + 2a-5z9 + 7a-4 - 25a-4z2 + 43a-4z4 - 34a-4z6 + 8a-4z8 - 6a-3z + 17a-3z3 - 9a-3z5 - 4a-3z7 + 2a-3z9 + 2a-2 - 7a-2z2 + 16a-2z4 - 13a-2z6 + 3a-2z8 - 2a-1z + 5a-1z3 - 4a-1z5 + a-1z7

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

{5, 9}

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 11173. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 15                 3 j = 13               2   j = 11             4 3   j = 9           4 2     j = 7         3 4       j = 5       4 4         j = 3     2 4           j = 1   1 3             j = -1   2               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, NonAlternating, 173]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 173]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[16, 6, 17, 5], X[12, 8, 13, 7], > X[20, 10, 21, 9], X[11, 3, 12, 2], X[18, 13, 19, 14], X[4, 16, 5, 15], > X[22, 17, 1, 18], X[8, 20, 9, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 173]]
Out[4]=	GaussCode[1, 6, -2, -8, 3, -1, 4, -10, 5, 2, -6, -4, 7, -11, 8, -3, 9, -7, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 173]]
Out[5]=	DTCode[6, -10, 16, 12, 20, -2, 18, 4, 22, 8, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 173]][t]
Out[6]=	-4 4 8 7 2 3 4 5 + t - -- + -- - - - 7 t + 8 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 173]][z]
Out[7]=	2 4 6 8 1 + 5 z + 4 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 173]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 173]], KnotSignature[Knot[11, NonAlternating, 173]]}
Out[9]=	{45, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 173]][q]
Out[10]=	1 2 3 4 5 6 7 3 - - - 5 q + 7 q - 7 q + 8 q - 6 q + 5 q - 3 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 173]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 173]][q]
Out[12]=	-2 2 8 10 14 16 20 22 26 28 1 - q - q + q + 4 q + 4 q - q - q - 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 173]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 6 -4 7 2 z 7 z 15 z 4 z 5 z 13 z 4 z z 6 z -- + -- - -- + -- - ---- + ----- - ---- - ---- + ----- - ---- - -- + ---- - 6 4 2 8 6 4 2 6 4 2 6 4 a a a a a a a a a a a a 6 8 z z > -- + -- 2 4 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 173]][a, z]
Out[14]=	2 2 2 2 4 7 2 3 z z 8 z 6 z 2 z 2 z 20 z 25 z 7 z -- + -- + -- + --- - -- - --- - --- - --- - ---- - ----- - ----- - ---- + 6 4 2 9 7 5 3 a 8 6 4 2 a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 5 8 z 20 z 17 z 5 z 3 z 30 z 43 z 16 z 9 z 14 z > ---- + ----- + ----- + ---- + ---- + ----- + ----- + ----- - ---- - ----- - 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a 5 5 6 6 6 7 7 7 7 8 9 z 4 z 21 z 34 z 13 z 3 z 2 z 4 z z 5 z > ---- - ---- - ----- - ----- - ----- + ---- - ---- - ---- + -- + ---- + 3 a 6 4 2 7 5 3 a 6 a a a a a a a a 8 8 9 9 8 z 3 z 2 z 2 z > ---- + ---- + ---- + ---- 4 2 5 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 173]], Vassiliev[3][Knot[11, NonAlternating, 173]]}
Out[15]=	{5, 9}
In[16]:=	Kh[Knot[11, NonAlternating, 173]][q, t]
Out[16]=	3 3 5 1 2 q 3 q 2 q 5 7 7 2 4 q + 4 q + ----- + ---- + -- + --- + ---- + 4 q t + 3 q t + 4 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 15 5 > 4 q t + 2 q t + 4 q t + 3 q t + 2 q t + 3 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n173

K11n172 K11n174