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

K11n173 K11n175

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X16,6,17,5 X14,7,15,8 X20,10,21,9 X11,5,12,4 X18,14,19,13 X2,16,3,15 X22,17,1,18 X8,20,9,19 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-3 + 11t-2 - 22t-1 + 27 - 22t + 11t2 - 2t3

Conway Polynomial:

1 + 4z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a64, ...}

Determinant and Signature:

{97, 4}

Jones Polynomial:

3q2 - 7q3 + 12q4 - 15q5 + 17q6 - 16q7 + 13q8 - 9q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

3q6 - 2q8 + 3q10 - 2q14 + 4q16 - 3q18 + 3q20 - 2q22 - q24 + 2q26 - 3q28 + 2q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 2a-8 + 6a-8z2 + 3a-8z4 - 3a-6 - 8a-6z2 - 7a-6z4 - 2a-6z6 + 3a-4 + 7a-4z2 + 3a-4z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + a-12z2 - 5a-12z4 + 4a-12z6 - 3a-11z + 8a-11z3 - 14a-11z5 + 8a-11z7 + a-10 - 3a-10z2 + 5a-10z4 - 12a-10z6 + 8a-10z8 - 3a-9z + 18a-9z3 - 27a-9z5 + 9a-9z7 + 3a-9z9 + 2a-8 - 9a-8z2 + 22a-8z4 - 28a-8z6 + 14a-8z8 - a-7z + 7a-7z3 - 12a-7z5 + 4a-7z7 + 3a-7z9 + 3a-6 - 14a-6z2 + 18a-6z4 - 12a-6z6 + 6a-6z8 - a-5z - 2a-5z3 + 3a-5z7 + 3a-4 - 9a-4z2 + 6a-4z4

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

{4, 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 11174. 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 j = 23                   1 j = 21                 3   j = 19               6 1   j = 17             7 3     j = 15           9 6       j = 13         8 7         j = 11       7 9           j = 9     5 8             j = 7   2 7               j = 5 1 5                 j = 3 3

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, 174]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 174]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[16, 6, 17, 5], X[14, 7, 15, 8], > X[20, 10, 21, 9], X[11, 5, 12, 4], X[18, 14, 19, 13], X[2, 16, 3, 15], > X[22, 17, 1, 18], X[8, 20, 9, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 174]]
Out[4]=	GaussCode[1, -8, -2, 6, 3, -1, 4, -10, 5, 2, -6, -11, 7, -4, 8, -3, 9, -7, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 174]]
Out[5]=	DTCode[6, -10, 16, 14, 20, -4, 18, 2, 22, 8, 12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 174]][t]
Out[6]=	2 11 22 2 3 27 - -- + -- - -- - 22 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 174]][z]
Out[7]=	2 4 6 1 + 4 z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 64], Knot[11, NonAlternating, 174]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 174]], KnotSignature[Knot[11, NonAlternating, 174]]}
Out[9]=	{97, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 174]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 3 q - 7 q + 12 q - 15 q + 17 q - 16 q + 13 q - 9 q + 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 174]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 174]][q]
Out[12]=	6 8 10 14 16 18 20 22 24 26 3 q - 2 q + 3 q - 2 q + 4 q - 3 q + 3 q - 2 q - q + 2 q - 28 30 34 > 3 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 174]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -10 2 3 3 z 6 z 8 z 7 z 3 z 7 z 3 z 2 z -a + -- - -- + -- - --- + ---- - ---- + ---- + ---- - ---- + ---- - ---- 8 6 4 10 8 6 4 8 6 4 6 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 174]][a, z]
Out[14]=	2 2 2 2 2 -10 2 3 3 3 z 3 z z z z 3 z 9 z 14 z 9 z a + -- + -- + -- - --- - --- - -- - -- + --- - ---- - ---- - ----- - ---- - 8 6 4 11 9 7 5 12 10 8 6 4 a a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 z 8 z 18 z 7 z 2 z 5 z 5 z 22 z 18 z 6 z > --- + ---- + ----- + ---- - ---- - ---- + ---- + ----- + ----- + ---- + 13 11 9 7 5 12 10 8 6 4 a a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 z 14 z 27 z 12 z 4 z 12 z 28 z 12 z 8 z 9 z > --- - ----- - ----- - ----- + ---- - ----- - ----- - ----- + ---- + ---- + 13 11 9 7 12 10 8 6 11 9 a a a a a a a a a a 7 7 8 8 8 9 9 4 z 3 z 8 z 14 z 6 z 3 z 3 z > ---- + ---- + ---- + ----- + ---- + ---- + ---- 7 5 10 8 6 9 7 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 174]], Vassiliev[3][Knot[11, NonAlternating, 174]]}
Out[15]=	{4, 9}
In[16]:=	Kh[Knot[11, NonAlternating, 174]][q, t]
Out[16]=	3 5 5 7 7 2 9 2 9 3 11 3 3 q + q + 5 q t + 2 q t + 7 q t + 5 q t + 8 q t + 7 q t + 11 4 13 4 13 5 15 5 15 6 17 6 > 9 q t + 8 q t + 7 q t + 9 q t + 6 q t + 7 q t + 17 7 19 7 19 8 21 8 23 9 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n174

K11n173 K11n175