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

K11n174 K11n176

Knotscape

This page is passe. Go here instead!    The Knot K11n175 Visit K11n175's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X5,17,6,16 X14,8,15,7 X9,21,10,20 X11,18,12,19 X13,3,14,2 X22,16,1,15 X17,12,18,13 X19,5,20,4 X21,9,22,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-3 + 9t-2 - 14t-1 + 15 - 14t + 9t2 - 2t3

Conway Polynomial:

1 + 4z2 - 3z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{65, 4}

Jones Polynomial:

1 - 3q + 6q2 - 8q3 + 11q4 - 11q5 + 10q6 - 8q7 + 5q8 - 2q9

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

{K11n103, ...}

A2 (sl(3)) Invariant:

1 - q2 + q6 - q8 + 4q10 + 2q14 - 3q18 + q20 - 2q22 + 2q24 + q26 - q28 - q32

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-11z + 3a-11z3 + a-10 - 4a-10z2 + 4a-10z4 + a-10z6 + 2a-9z3 - 2a-9z5 + 3a-9z7 + 2a-8 - 10a-8z2 + 15a-8z4 - 9a-8z6 + 4a-8z8 + 2a-7z - 5a-7z3 + a-7z5 - a-7z7 + 2a-7z9 + 3a-6 - 15a-6z2 + 25a-6z4 - 23a-6z6 + 8a-6z8 + 2a-5z3 - 6a-5z5 - a-5z7 + 2a-5z9 + 3a-4 - 7a-4z2 + 11a-4z4 - 12a-4z6 + 4a-4z8 + 6a-3z3 - 9a-3z5 + 3a-3z7 + 2a-2z2 - 3a-2z4 + a-2z6

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 11175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                   2 j = 17                 3   j = 15               5 2   j = 13             5 3     j = 11           6 5       j = 9         5 5         j = 7       3 6           j = 5     3 5             j = 3   1 4               j = 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, NonAlternating, 175]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 175]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[5, 17, 6, 16], X[14, 8, 15, 7], > X[9, 21, 10, 20], X[11, 18, 12, 19], X[13, 3, 14, 2], X[22, 16, 1, 15], > X[17, 12, 18, 13], X[19, 5, 20, 4], X[21, 9, 22, 8]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 175]]
Out[4]=	GaussCode[1, 7, -2, 10, -3, -1, 4, 11, -5, 2, -6, 9, -7, -4, 8, 3, -9, 6, -10, > 5, -11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 175]]
Out[5]=	DTCode[6, -10, -16, 14, -20, -18, -2, 22, -12, -4, -8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 175]][t]
Out[6]=	2 9 14 2 3 15 - -- + -- - -- - 14 t + 9 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 175]][z]
Out[7]=	2 4 6 1 + 4 z - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 175]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 175]], KnotSignature[Knot[11, NonAlternating, 175]]}
Out[9]=	{65, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 175]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 3 q + 6 q - 8 q + 11 q - 11 q + 10 q - 8 q + 5 q - 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 103], Knot[11, NonAlternating, 175]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 175]][q]
Out[12]=	2 6 8 10 14 18 20 22 24 26 28 32 1 - q + q - q + 4 q + 2 q - 3 q + q - 2 q + 2 q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 175]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 6 -10 2 3 3 3 z 3 z 2 z 2 z z 3 z 2 z z z -a + -- - -- + -- + ---- - ---- + ---- + ---- + -- - ---- - ---- + -- - -- - 8 6 4 8 6 4 2 8 6 4 2 6 a a a a a a a a a a a a 6 z > -- 4 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 175]][a, z]
Out[14]=	2 2 2 2 2 3 -10 2 3 3 2 z 2 z 4 z 10 z 15 z 7 z 2 z 3 z a + -- + -- + -- - --- + --- - ---- - ----- - ----- - ---- + ---- + ---- + 8 6 4 11 7 10 8 6 4 2 11 a a a a a a a a a a a 3 3 3 3 4 4 4 4 4 5 2 z 5 z 2 z 6 z 4 z 15 z 25 z 11 z 3 z 2 z > ---- - ---- + ---- + ---- + ---- + ----- + ----- + ----- - ---- - ---- + 9 7 5 3 10 8 6 4 2 9 a a a a a a a a a a 5 5 5 6 6 6 6 6 7 7 7 z 6 z 9 z z 9 z 23 z 12 z z 3 z z z > -- - ---- - ---- + --- - ---- - ----- - ----- + -- + ---- - -- - -- + 7 5 3 10 8 6 4 2 9 7 5 a a a a a a a a a a a 7 8 8 8 9 9 3 z 4 z 8 z 4 z 2 z 2 z > ---- + ---- + ---- + ---- + ---- + ---- 3 8 6 4 7 5 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 175]], Vassiliev[3][Knot[11, NonAlternating, 175]]}
Out[15]=	{4, 9}
In[16]:=	Kh[Knot[11, NonAlternating, 175]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 9 3 4 q + 3 q + ---- + --- + -- + 5 q t + 3 q t + 6 q t + 5 q t + 5 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 6 q t + 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 17 6 19 7 > 3 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n175

K11n174 K11n176