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

K11a273 K11a275

Knotscape

This page is passe. Go here instead!    The Knot K11a274 Visit K11a274's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 14 18 2 20 22 4 12 8 16

Alexander Polynomial:

t-4 - 6t-3 + 18t-2 - 35t-1 + 45 - 35t + 18t2 - 6t3 + t4

Conway Polynomial:

1 - z2 + 2z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{165, 0}

Jones Polynomial:

q-6 - 4q-5 + 10q-4 - 17q-3 + 23q-2 - 27q-1 + 27 - 23q + 18q2 - 10q3 + 4q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-18 - q-16 + q-14 + 3q-12 - 5q-10 + 3q-8 - 3q-6 - 2q-4 + 4q-2 - 4 + 7q2 - 3q4 + 2q6 + 3q8 - 4q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- a-2 - 4a-2z2 - 3a-2z4 - a-2z6 + 5 + 11z2 + 11z4 + 5z6 + z8 - 5a2 - 10a2z2 - 7a2z4 - 2a2z6 + 2a4 + 2a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 4a-4z4 + 4a-4z6 - a-3z + 7a-3z3 - 12a-3z5 + 9a-3z7 + a-2 - 9a-2z2 + 19a-2z4 - 22a-2z6 + 13a-2z8 - a-1z + 8a-1z3 - 8a-1z5 - 7a-1z7 + 10a-1z9 + 5 - 27z2 + 56z4 - 59z6 + 21z8 + 3z10 + az - az3 + 10az5 - 30az7 + 18az9 + 5a2 - 24a2z2 + 48a2z4 - 51a2z6 + 16a2z8 + 3a2z10 + 4a3z3 - 3a3z5 - 10a3z7 + 8a3z9 + 2a4 - 6a4z2 + 13a4z4 - 17a4z6 + 8a4z8 - a5z + 5a5z3 - 8a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

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

{-1, 2}

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=0 is the signature of 11274. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   7 1   j = 5                 11 3     j = 3               12 7       j = 1             15 11         j = -1           13 13           j = -3         10 14             j = -5       7 13               j = -7     3 10                 j = -9   1 7                   j = -11   3                     j = -13 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, 274]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 274]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 5, 15, 6], X[18, 7, 19, 8], > X[2, 10, 3, 9], X[20, 11, 21, 12], X[22, 14, 1, 13], X[4, 15, 5, 16], > X[12, 17, 13, 18], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 274]]
Out[4]=	GaussCode[1, -5, 2, -8, 3, -1, 4, -10, 5, -2, 6, -9, 7, -3, 8, -11, 9, -4, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 274]]
Out[5]=	DTCode[6, 10, 14, 18, 2, 20, 22, 4, 12, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 274]][t]
Out[6]=	-4 6 18 35 2 3 4 45 + t - -- + -- - -- - 35 t + 18 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 274]][z]
Out[7]=	2 4 6 8 1 - z + 2 z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 274]}
In[9]:=	{KnotDet[Knot[11, Alternating, 274]], KnotSignature[Knot[11, Alternating, 274]]}
Out[9]=	{165, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 274]][q]
Out[10]=	-6 4 10 17 23 27 2 3 4 5 27 + q - -- + -- - -- + -- - -- - 23 q + 18 q - 10 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 274]}
In[12]:=	A2Invariant[Knot[11, Alternating, 274]][q]
Out[12]=	-18 -16 -14 3 5 3 3 2 4 2 4 6 -4 + q - q + q + --- - --- + -- - -- - -- + -- + 7 q - 3 q + 2 q + 12 10 8 6 4 2 q q q q q q 8 10 12 14 > 3 q - 4 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 274]][a, z]
Out[13]=	2 4 -2 2 4 2 4 z 2 2 4 2 4 3 z 5 - a - 5 a + 2 a + 11 z - ---- - 10 a z + 2 a z + 11 z - ---- - 2 2 a a 6 2 4 4 4 6 z 2 6 8 > 7 a z + a z + 5 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 274]][a, z]
Out[14]=	2 2 -2 2 4 z z 5 2 z 9 z 2 2 5 + a + 5 a + 2 a - -- - - + a z - a z - 27 z + -- - ---- - 24 a z - 3 a 4 2 a a a 3 3 3 4 2 6 2 z 7 z 8 z 3 3 3 5 3 4 > 6 a z + a z - -- + ---- + ---- - a z + 4 a z + 5 a z + 56 z - 5 3 a a a 4 4 5 5 5 4 z 19 z 2 4 4 4 6 4 z 12 z 8 z > ---- + ----- + 48 a z + 13 a z - 2 a z + -- - ----- - ---- + 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 4 z 22 z 2 6 4 6 > 10 a z - 3 a z - 8 a z - 59 z + ---- - ----- - 51 a z - 17 a z + 4 2 a a 7 7 8 6 6 9 z 7 z 7 3 7 5 7 8 13 z > a z + ---- - ---- - 30 a z - 10 a z + 4 a z + 21 z + ----- + 3 a 2 a a 9 2 8 4 8 10 z 9 3 9 10 2 10 > 16 a z + 8 a z + ----- + 18 a z + 8 a z + 3 z + 3 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 274]], Vassiliev[3][Knot[11, Alternating, 274]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 274]][q, t]
Out[16]=	13 1 3 1 7 3 10 7 13 -- + 15 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 10 14 13 3 3 2 5 2 5 3 > ----- + ---- + --- + 11 q t + 12 q t + 7 q t + 11 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a274

K11a273 K11a275