10 153: Difference between revisions

Revision as of 19:14, 28 August 2005

10_152

10_154

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Visit 10 153's page at Knotilus!

Visit 10 153's page at the original Knot Atlas!

10 153 Quick Notes

10_153 is not $k$ -colourable for any $k$ . See The Determinant and the Signature.

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_12,6,13,5 X_13,18,14,19 X_9,16,10,17 X_17,10,18,11 X_15,20,16,1 X_19,14,20,15 X_6,12,7,11 X₂₈₃₇
Gauss code	1, -10, 2, -1, 3, -9, 10, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7
Dowker-Thistlethwaite code	4 8 12 2 -16 6 -18 -20 -10 -14
Conway Notation	[(3,2)-(21,2)]

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-4]
Hyperbolic Volume	7.37434
A-Polynomial	See Data:10 153/A-polynomial

[edit Notes for 10 153's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$0$
Topological 4 genus	$0$
Concordance genus	$0$
Rasmussen s-Invariant	0

[edit Notes for 10 153's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-t^{2}-t+3-t^{-1}-t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+5z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 1, 0 }
Jones polynomial	$-q^{4}+q^{3}-q^{2}+q+1+q^{-2}-q^{-3}+q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-z^{4}a^{-2}+6z^{4}-a^{4}z^{2}-a^{2}z^{2}-4z^{2}a^{-2}+10z^{2}-a^{4}-a^{2}-3a^{-2}+6$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}+az^{7}+2z^{7}a^{-1}+z^{7}a^{-3}+a^{4}z^{6}-6z^{6}a^{-2}-7z^{6}+a^{5}z^{5}+a^{3}z^{5}-7az^{5}-13z^{5}a^{-1}-6z^{5}a^{-3}-4a^{4}z^{4}+10z^{4}a^{-2}+14z^{4}-4a^{5}z^{3}-4a^{3}z^{3}+12az^{3}+22z^{3}a^{-1}+10z^{3}a^{-3}+3a^{4}z^{2}-2a^{2}z^{2}-7z^{2}a^{-2}-12z^{2}+3a^{5}z+2a^{3}z-6az-10za^{-1}-5za^{-3}-a^{4}+a^{2}+3a^{-2}+6$
The A2 invariant	$-q^{16}-q^{12}-q^{10}+2q^{4}+2q^{2}+3+2q^{-2}-q^{-8}-q^{-10}-q^{-12}$
The G2 invariant	$q^{80}+q^{76}-q^{74}+q^{70}-2q^{68}+q^{64}-3q^{62}+q^{60}-q^{58}-4q^{56}+5q^{54}-5q^{52}+q^{50}+q^{48}-5q^{46}+5q^{44}-3q^{42}-2q^{40}+2q^{38}-4q^{36}+q^{34}+3q^{32}-5q^{30}+3q^{28}-q^{24}+2q^{22}-2q^{20}+2q^{18}+4q^{14}-2q^{12}+3q^{10}+4q^{8}-q^{6}+5q^{4}-q^{2}+2+6q^{-2}-q^{-4}+2q^{-6}+4q^{-8}-2q^{-10}+6q^{-12}-q^{-14}-3q^{-16}+4q^{-18}-4q^{-20}+2q^{-22}-3q^{-26}+q^{-28}-q^{-30}-3q^{-32}-2q^{-36}-q^{-38}-3q^{-42}-q^{-48}-q^{-52}+q^{-56}+q^{-60}$

Further Quantum Invariants

Further quantum knot invariants for 10_153.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+q^{3}+q+2q^{-1}-q^{-9}$
2	$q^{32}-q^{28}+q^{24}-3q^{20}-q^{18}+2q^{16}-2q^{14}-q^{12}+2q^{10}+q^{6}+q^{4}+2q^{2}+1+q^{-2}+2q^{-4}-2q^{-8}+q^{-10}+q^{-12}-2q^{-14}-q^{-16}-q^{-24}+q^{-28}$
3	$-q^{63}+q^{59}+q^{57}-2q^{53}-q^{51}+q^{49}+4q^{47}+3q^{45}-2q^{43}-5q^{41}-2q^{39}+5q^{37}+4q^{35}-4q^{33}-7q^{31}-q^{29}+5q^{27}+q^{25}-4q^{23}-2q^{21}+q^{19}+q^{17}+q^{15}+q^{9}+q^{7}-q^{5}+q^{3}+5q+4q^{-1}-2q^{-3}-3q^{-5}+6q^{-7}+5q^{-9}-3q^{-11}-7q^{-13}-q^{-15}+5q^{-17}+2q^{-19}-3q^{-21}-5q^{-23}-2q^{-25}+3q^{-27}+3q^{-29}-3q^{-33}-2q^{-35}+q^{-37}+q^{-39}+q^{-41}-q^{-47}+q^{-51}+q^{-53}-q^{-57}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}-q^{12}-q^{10}+2q^{4}+2q^{2}+3+2q^{-2}-q^{-8}-q^{-10}-q^{-12}$
1,1	$q^{44}+2q^{40}-2q^{38}+2q^{34}-4q^{32}+6q^{30}-8q^{28}+4q^{26}-2q^{24}-4q^{22}+2q^{20}-12q^{18}+4q^{16}-6q^{14}+3q^{12}+6q^{8}+4q^{6}+5q^{4}+8q^{2}+10q^{-2}-2q^{-4}+4q^{-6}-4q^{-10}+4q^{-12}-6q^{-14}-2q^{-16}-2q^{-18}-4q^{-20}+2q^{-22}-4q^{-24}+2q^{-32}+q^{-36}$
2,0	$q^{42}+q^{36}+q^{34}+q^{32}-q^{28}-2q^{26}-5q^{24}-2q^{22}-3q^{20}-3q^{18}-q^{16}+q^{12}+3q^{8}+3q^{6}+6q^{4}+5q^{2}+8+4q^{-2}+2q^{-4}+q^{-6}-2q^{-8}-2q^{-10}-2q^{-12}-2q^{-14}-2q^{-16}-3q^{-18}-2q^{-20}-q^{-22}-q^{-24}+q^{-30}+q^{-32}+q^{-34}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}+q^{30}-q^{24}-3q^{22}-q^{20}-3q^{18}-3q^{16}-q^{14}-2q^{12}-2q^{10}+q^{8}+3q^{6}+5q^{4}+8q^{2}+10+7q^{-2}+4q^{-4}-q^{-6}-2q^{-8}-7q^{-10}-5q^{-12}-3q^{-14}-3q^{-16}+q^{-20}+q^{-22}+q^{-26}$
1,0,0	$-q^{21}-q^{17}-q^{15}-q^{13}-q^{11}+2q^{5}+3q^{3}+4q+3q^{-1}+3q^{-3}-q^{-7}-q^{-9}-2q^{-11}-q^{-13}-q^{-15}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}+q^{40}+2q^{38}+q^{36}+q^{34}+q^{32}-q^{30}-3q^{28}-3q^{26}-5q^{24}-6q^{22}-7q^{20}-6q^{18}-7q^{16}-8q^{14}-2q^{12}+q^{10}+4q^{8}+13q^{6}+19q^{4}+20q^{2}+20+17q^{-2}+8q^{-4}-8q^{-8}-11q^{-10}-15q^{-12}-13q^{-14}-8q^{-16}-5q^{-18}-2q^{-20}+2q^{-22}+3q^{-24}+2q^{-26}+2q^{-28}+q^{-30}+q^{-32}$
1,0,0,0	$-q^{26}-q^{22}-q^{20}-q^{18}-q^{16}-q^{14}-q^{12}+2q^{6}+3q^{4}+5q^{2}+4+4q^{-2}+3q^{-4}-q^{-8}-2q^{-10}-2q^{-12}-2q^{-14}-q^{-16}-q^{-18}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}-q^{30}-q^{24}+q^{22}-q^{20}+q^{18}-q^{16}+q^{14}+q^{8}-q^{6}+3q^{4}+2+q^{-2}+2q^{-4}+q^{-6}+q^{-10}-q^{-12}+q^{-14}-q^{-16}-q^{-20}-q^{-22}-q^{-26}$
1,0	$q^{56}+q^{48}-q^{44}+q^{40}-q^{38}-2q^{36}-q^{34}-q^{30}-3q^{28}-2q^{26}-2q^{20}-q^{18}+q^{14}+q^{12}+q^{10}+2q^{8}+4q^{6}+3q^{4}+4q^{2}+3+4q^{-2}+3q^{-4}+3q^{-6}-q^{-8}-q^{-14}-3q^{-16}-3q^{-18}-q^{-20}-q^{-22}-3q^{-24}-2q^{-26}+q^{-36}+q^{-44}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}+q^{42}+q^{38}-2q^{34}-q^{32}-3q^{30}-q^{28}-3q^{26}-q^{24}-3q^{22}-q^{20}-q^{18}-2q^{16}-2q^{14}-2q^{12}+q^{10}+q^{8}+6q^{6}+6q^{4}+10q^{2}+10+10q^{-2}+6q^{-4}+4q^{-6}-4q^{-10}-5q^{-12}-7q^{-14}-5q^{-16}-6q^{-18}-2q^{-20}-2q^{-22}+q^{-26}+q^{-28}+q^{-30}+q^{-34}$

G2 Invariants.

Weight

Invariant

1,0

q^{80}+q^{76}-q^{74}+q^{70}-2q^{68}+q^{64}-3q^{62}+q^{60}-q^{58}-4q^{56}+5q^{54}-5q^{52}+q^{50}+q^{48}-5q^{46}+5q^{44}-3q^{42}-2q^{40}+2q^{38}-4q^{36}+q^{34}+3q^{32}-5q^{30}+3q^{28}-q^{24}+2q^{22}-2q^{20}+2q^{18}+4q^{14}-2q^{12}+3q^{10}+4q^{8}-q^{6}+5q^{4}-q^{2}+2+6q^{-2}-q^{-4}+2q^{-6}+4q^{-8}-2q^{-10}+6q^{-12}-q^{-14}-3q^{-16}+4q^{-18}-4q^{-20}+2q^{-22}-3q^{-26}+q^{-28}-q^{-30}-3q^{-32}-2q^{-36}-q^{-38}-3q^{-42}-q^{-48}-q^{-52}+q^{-56}+q^{-60}

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 153"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= $t^{3}-t^{2}-t+3-t^{-1}-t^{-2}+t^{-3}$

In[5]:= Conway[K][z]

Out[5]= $z^{6}+5z^{4}+4z^{2}+1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= $\{1\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 1, 0 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= $-q^{4}+q^{3}-q^{2}+q+1+q^{-2}-q^{-3}+q^{-4}-q^{-5}$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $z^{6}-z^{4}a^{-2}+6z^{4}-a^{4}z^{2}-a^{2}z^{2}-4z^{2}a^{-2}+10z^{2}-a^{4}-a^{2}-3a^{-2}+6$

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= $z^{8}a^{-2}+z^{8}+az^{7}+2z^{7}a^{-1}+z^{7}a^{-3}+a^{4}z^{6}-6z^{6}a^{-2}-7z^{6}+a^{5}z^{5}+a^{3}z^{5}-7az^{5}-13z^{5}a^{-1}-6z^{5}a^{-3}-4a^{4}z^{4}+10z^{4}a^{-2}+14z^{4}-4a^{5}z^{3}-4a^{3}z^{3}+12az^{3}+22z^{3}a^{-1}+10z^{3}a^{-3}+3a^{4}z^{2}-2a^{2}z^{2}-7z^{2}a^{-2}-12z^{2}+3a^{5}z+2a^{3}z-6az-10za^{-1}-5za^{-3}-a^{4}+a^{2}+3a^{-2}+6$

Vassiliev invariants

V₂ and V₃:

(4, -1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

16

-8

128

{\frac {488}{3}}

{\frac {64}{3}}

-128

-{\frac {560}{3}}

-{\frac {128}{3}}

-8

{\frac {2048}{3}}

32

{\frac {7808}{3}}

{\frac {1024}{3}}

{\frac {41222}{15}}

{\frac {464}{5}}

{\frac {44528}{45}}

{\frac {442}{9}}

{\frac {1622}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

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

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

9

1

-1

7

0

5

1

0

3

1

0

1

2

-1

1

3

1

-3

1

-5

1

0

-7

1

0

-9

0

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$	$i=1$
$r=-5$		${\mathbb {Z} }$
$r=-4$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$		${\mathbb {Z} }$
$r=-2$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }$
$r=1$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=4$	${\mathbb {Z} }$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 153]]
Out[2]=	10
In[3]:=	PD[Knot[10, 153]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[13, 18, 14, 19], X[9, 16, 10, 17], X[17, 10, 18, 11], X[15, 20, 16, 1], X[19, 14, 20, 15], X[6, 12, 7, 11], X[2, 8, 3, 7]]
In[4]:=	GaussCode[Knot[10, 153]]
Out[4]=	GaussCode[1, -10, 2, -1, 3, -9, 10, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7]
In[5]:=	BR[Knot[10, 153]]
Out[5]=	BR[4, {-1, -1, -1, -2, -1, -1, 3, 2, 2, 2, 3}]
In[6]:=	alex = Alexander[Knot[10, 153]][t]
Out[6]=	-3 -2 1 2 3 3 + t - t - - - t - t + t t
In[7]:=	Conway[Knot[10, 153]][z]
Out[7]=	2 4 6 1 + 4 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 153]}
In[9]:=	{KnotDet[Knot[10, 153]], KnotSignature[Knot[10, 153]]}
Out[9]=	{1, 0}
In[10]:=	J=Jones[Knot[10, 153]][q]
Out[10]=	-5 -4 -3 -2 2 3 4 1 - q + q - q + q + q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 153]}
In[12]:=	A2Invariant[Knot[10, 153]][q]
Out[12]=	-16 -12 -10 2 2 2 8 10 12 3 - q - q - q + -- + -- + 2 q - q - q - q 4 2 q q
In[13]:=	Kauffman[Knot[10, 153]][a, z]
Out[13]=	3 2 4 5 z 10 z 3 5 2 6 + -- + a - a - --- - ---- - 6 a z + 2 a z + 3 a z - 12 z - 2 3 a a a 2 3 3 7 z 2 2 4 2 10 z 22 z 3 3 3 ---- - 2 a z + 3 a z + ----- + ----- + 12 a z - 4 a z - 2 3 a a a 4 5 5 5 3 4 10 z 4 4 6 z 13 z 5 3 5 4 a z + 14 z + ----- - 4 a z - ---- - ----- - 7 a z + a z + 2 3 a a a 6 7 7 8 5 5 6 6 z 4 6 z 2 z 7 8 z a z - 7 z - ---- + a z + -- + ---- + a z + z + -- 2 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[10, 153]], Vassiliev[3][Knot[10, 153]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[10, 153]][q, t]
Out[15]=	3 1 1 1 1 1 1 1 t - + q + ------ + ----- + ----- + ----- + ----- + ---- + --- + - + q 11 5 7 4 7 3 5 2 3 2 5 q t q q t q t q t q t q t q t 3 2 3 2 5 3 5 4 9 5 q t + q t + q t + q t + q t + q t

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+<!-- -->
+<!--  -->
+<!-- -->
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+<!-- -->
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 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=10|k=153|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,3,-9,10,-2,-5,6,9,-3,-4,8,-7,5,-6,4,-8,7/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=10|k=153|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,3,-9,10,-2,-5,6,9,-3,-4,8,-7,5,-6,4,-8,7/goTop.html}}
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 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=13.3333%><table cellpadding=0 cellspacing=0>
@@ Line 47: / Line 40: @@
 <tr align=center><td>-9</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
 <tr align=center><td>-11</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 127: / Line 119: @@
   q  t + q t  + q  t  + q  t  + q  t  + q  t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

10 153: Difference between revisions

Revision as of 19:14, 28 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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