10 145

10_144

10_146

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10 145 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X5,12,6,13 X8394 X2,9,3,10 X11,16,12,17 X17,10,18,11 X7,18,8,19 X13,20,14,1 X19,14,20,15 X15,6,16,7
Gauss code	1, -4, 3, -1, -2, 10, -7, -3, 4, 6, -5, 2, -8, 9, -10, 5, -6, 7, -9, 8
Dowker-Thistlethwaite code	4 8 -12 -18 2 -16 -20 -6 -10 -14
Conway Notation	[22,3,3-]

Minimum Braid Representative: Length is 11, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-12][3] Hyperbolic Volume 5.0449 A-Polynomial See Data:10 145/A-polynomial

[edit Notes for 10 145's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $[1,2]$ Concordance genus $2$ Rasmussen s-Invariant 4

[edit Notes for 10 145's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}+t-3+t^{-1}+t^{-2}$
Conway polynomial	$z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 3, -2 }
Jones polynomial	$q^{-2}+q^{-7}-q^{-8}+q^{-9}-q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$-a^{10}+z^{2}a^{8}+a^{8}-a^{6}+z^{4}a^{4}+4z^{2}a^{4}+2a^{4}$
Kauffman polynomial (db, data sources)	$z^{7}a^{11}-6z^{5}a^{11}+10z^{3}a^{11}-5za^{11}+z^{8}a^{10}-6z^{6}a^{10}+10z^{4}a^{10}-6z^{2}a^{10}+a^{10}+2z^{7}a^{9}-12z^{5}a^{9}+18z^{3}a^{9}-6za^{9}+z^{8}a^{8}-6z^{6}a^{8}+9z^{4}a^{8}-4z^{2}a^{8}+a^{8}+z^{7}a^{7}-6z^{5}a^{7}+8z^{3}a^{7}-2za^{7}-2z^{2}a^{6}+a^{6}-za^{5}+z^{4}a^{4}-4z^{2}a^{4}+2a^{4}$
The A2 invariant	$-q^{32}-q^{30}+q^{24}+q^{14}+q^{10}+q^{8}+q^{6}$
The G2 invariant	$q^{156}+q^{152}-q^{148}+q^{142}-2q^{138}-q^{130}-3q^{128}-q^{126}+q^{124}-q^{122}-q^{120}-q^{118}-2q^{116}+2q^{114}-2q^{112}-q^{110}+q^{108}+2q^{104}+2q^{102}+q^{98}+q^{96}+2q^{92}+q^{88}+2q^{82}-q^{78}-3q^{76}+q^{74}-q^{72}-3q^{66}+2q^{64}+q^{62}-2q^{60}-q^{54}+3q^{52}+2q^{48}+q^{46}+3q^{42}+q^{38}+q^{32}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 10_145.

A1 Invariants.

Weight	Invariant
1	$-q^{21}+q^{13}+q^{5}+q^{3}$
2	$q^{60}-q^{56}-q^{46}+q^{44}+q^{42}-q^{40}-q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+q^{24}+2q^{22}-q^{18}+2q^{16}-q^{12}+q^{10}+q^{8}+q^{6}$
3	$-q^{117}+q^{113}+q^{111}-q^{107}+q^{97}-q^{95}-2q^{93}+2q^{89}+2q^{87}-q^{85}-q^{83}+q^{79}+2q^{77}-2q^{73}-q^{71}+q^{69}+q^{67}-2q^{65}-q^{63}+q^{61}-q^{57}-q^{55}+q^{53}-q^{49}-q^{47}-q^{41}-q^{39}+2q^{35}+2q^{33}-q^{29}+2q^{25}+2q^{23}-q^{19}-q^{17}+q^{15}+q^{13}+q^{11}+q^{9}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{32}-q^{30}+q^{24}+q^{14}+q^{10}+q^{8}+q^{6}$
1,1	$q^{84}+2q^{80}-2q^{74}-4q^{72}-2q^{68}+2q^{66}+2q^{64}+2q^{62}+6q^{60}-2q^{58}-4q^{54}-3q^{52}-2q^{50}-2q^{48}+2q^{46}-2q^{44}+2q^{42}-4q^{40}+2q^{38}-4q^{36}+2q^{34}+2q^{32}-2q^{30}+4q^{28}+4q^{24}+q^{20}+2q^{18}+2q^{16}+2q^{14}+q^{12}$
2,0	$q^{82}+q^{80}+q^{78}-q^{76}-q^{74}-q^{72}-q^{70}+q^{64}+q^{60}+q^{58}-q^{56}-q^{54}-q^{52}-2q^{48}-2q^{46}-q^{44}-2q^{42}-q^{40}+2q^{36}+q^{34}+2q^{32}+2q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+2q^{16}+q^{14}+q^{12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{66}+q^{62}+q^{60}-2q^{56}-q^{54}-2q^{52}-3q^{50}-q^{46}+2q^{40}+q^{38}+q^{34}-q^{30}-q^{28}+q^{26}+q^{22}+3q^{20}+q^{18}+2q^{16}+q^{14}+q^{12}$
1,0,0	$-q^{43}-q^{41}-q^{39}+q^{33}+q^{31}-q^{25}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{88}+q^{86}+q^{84}+q^{82}+q^{80}-q^{74}-2q^{72}-2q^{70}-2q^{68}-3q^{66}-3q^{64}-2q^{62}-2q^{60}-2q^{58}-q^{56}+2q^{54}+2q^{52}+3q^{50}+4q^{48}+2q^{46}-q^{42}-2q^{40}-3q^{38}-q^{36}+q^{34}+2q^{32}+3q^{30}+3q^{28}+4q^{26}+2q^{24}+2q^{22}+q^{20}+q^{18}$
1,0,0,0	$-q^{54}-q^{52}-q^{50}-q^{48}+q^{42}+q^{40}+q^{38}-q^{32}-q^{30}+q^{24}+q^{22}+2q^{20}+q^{18}+q^{16}+q^{14}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{66}-q^{62}-q^{60}+q^{54}+q^{50}+q^{46}-q^{38}-q^{34}-q^{30}+q^{28}+q^{26}+q^{22}+q^{20}+q^{18}+q^{14}+q^{12}$
1,0	$q^{108}+q^{100}-q^{90}-2q^{88}-q^{82}-q^{80}-q^{72}+q^{56}+q^{48}-q^{44}+q^{42}+q^{40}-q^{36}+q^{34}+q^{32}+q^{30}+q^{26}+q^{24}+q^{22}+q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{90}+q^{86}+q^{84}+q^{82}-q^{78}-q^{76}-3q^{74}-2q^{72}-3q^{70}-2q^{68}-2q^{66}+q^{60}+2q^{58}+2q^{56}+2q^{54}+q^{52}+q^{50}-2q^{44}-q^{42}-2q^{40}+2q^{32}+2q^{30}+3q^{28}+2q^{26}+2q^{24}+q^{22}+q^{20}+q^{18}$

G2 Invariants.

Weight	Invariant
1,0	$q^{156}+q^{152}-q^{148}+q^{142}-2q^{138}-q^{130}-3q^{128}-q^{126}+q^{124}-q^{122}-q^{120}-q^{118}-2q^{116}+2q^{114}-2q^{112}-q^{110}+q^{108}+2q^{104}+2q^{102}+q^{98}+q^{96}+2q^{92}+q^{88}+2q^{82}-q^{78}-3q^{76}+q^{74}-q^{72}-3q^{66}+2q^{64}+q^{62}-2q^{60}-q^{54}+3q^{52}+2q^{48}+q^{46}+3q^{42}+q^{38}+q^{32}+q^{30}$

.

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 145"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{2}+t-3+t^{-1}+t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}+5z^{2}+1$

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]=

{ 3, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-2}+q^{-7}-q^{-8}+q^{-9}-q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{10}+z^{2}a^{8}+a^{8}-a^{6}+z^{4}a^{4}+4z^{2}a^{4}+2a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{11}-6z^{5}a^{11}+10z^{3}a^{11}-5za^{11}+z^{8}a^{10}-6z^{6}a^{10}+10z^{4}a^{10}-6z^{2}a^{10}+a^{10}+2z^{7}a^{9}-12z^{5}a^{9}+18z^{3}a^{9}-6za^{9}+z^{8}a^{8}-6z^{6}a^{8}+9z^{4}a^{8}-4z^{2}a^{8}+a^{8}+z^{7}a^{7}-6z^{5}a^{7}+8z^{3}a^{7}-2za^{7}-2z^{2}a^{6}+a^{6}-za^{5}+z^{4}a^{4}-4z^{2}a^{4}+2a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {...}

Vassiliev invariants

V2 and V3:

(5, -12)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $20$ $-96$ $200$ ${\frac {1702}{3}}$ ${\frac {266}{3}}$ $-1920$ $-3712$ $-640$ $-544$ ${\frac {4000}{3}}$ $4608$ ${\frac {34040}{3}}$ ${\frac {5320}{3}}$ ${\frac {147487}{6}}$ $38$ ${\frac {93134}{9}}$ ${\frac {4837}{18}}$ ${\frac {8287}{6}}$

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 $s=$-2 is the signature of 10 145. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                   1 1 -5                   1 1 -7             1 1     0 -9                     0 -11         1 2 1       0 -13       1             1 -15       1 1           0 -17   1 1               0 -19                     0 -21 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{-4}-q^{-7}+q^{-8}+2q^{-9}-4q^{-10}+2q^{-11}+4q^{-12}-5q^{-13}+2q^{-14}+2q^{-15}-5q^{-16}+2q^{-17}+2q^{-18}-4q^{-19}+2q^{-20}+q^{-21}-2q^{-22}+2q^{-23}-q^{-24}-q^{-25}+2q^{-26}-q^{-27}-q^{-28}+q^{-29}$
3	$q^{-6}-q^{-10}+q^{-12}+2q^{-13}-q^{-14}-2q^{-15}+3q^{-17}+q^{-18}-2q^{-19}-2q^{-20}+2q^{-21}+q^{-22}-q^{-23}-2q^{-24}+q^{-25}+q^{-26}-q^{-28}-q^{-29}+q^{-30}+q^{-31}-3q^{-33}+4q^{-35}-5q^{-37}-q^{-38}+6q^{-39}+2q^{-40}-6q^{-41}-2q^{-42}+5q^{-43}+2q^{-44}-3q^{-45}-2q^{-46}+3q^{-47}-2q^{-49}+2q^{-51}-2q^{-53}+q^{-55}+q^{-56}-q^{-57}$
4	$q^{-8}-q^{-13}+q^{-16}+2q^{-17}-q^{-18}+2q^{-19}-4q^{-20}-q^{-21}+q^{-22}+9q^{-24}-3q^{-25}-4q^{-26}-8q^{-27}-2q^{-28}+17q^{-29}+2q^{-30}-4q^{-31}-14q^{-32}-5q^{-33}+20q^{-34}+3q^{-35}-2q^{-36}-16q^{-37}-6q^{-38}+21q^{-39}+3q^{-40}-4q^{-41}-16q^{-42}-5q^{-43}+21q^{-44}+4q^{-45}-5q^{-46}-14q^{-47}-5q^{-48}+18q^{-49}+5q^{-50}-3q^{-51}-12q^{-52}-6q^{-53}+13q^{-54}+7q^{-55}-q^{-56}-10q^{-57}-6q^{-58}+8q^{-59}+6q^{-60}-q^{-61}-5q^{-62}-3q^{-63}+4q^{-64}+3q^{-65}-4q^{-66}-2q^{-67}+q^{-68}+6q^{-69}+2q^{-70}-8q^{-71}-3q^{-72}+q^{-73}+7q^{-74}+3q^{-75}-5q^{-76}-3q^{-77}-2q^{-78}+5q^{-79}+q^{-80}-2q^{-81}-q^{-82}-q^{-83}+4q^{-84}-q^{-85}-q^{-86}-q^{-87}-q^{-88}+3q^{-89}-q^{-92}-q^{-93}+q^{-94}$
5	Not Available
6	Not Available
7	Not Available

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 145]]
Out[2]=	PD[X[4, 2, 5, 1], X[5, 12, 6, 13], X[8, 3, 9, 4], X[2, 9, 3, 10], X[11, 16, 12, 17], X[17, 10, 18, 11], X[7, 18, 8, 19], X[13, 20, 14, 1], X[19, 14, 20, 15], X[15, 6, 16, 7]]
In[3]:=	GaussCode[Knot[10, 145]]
Out[3]=	GaussCode[1, -4, 3, -1, -2, 10, -7, -3, 4, 6, -5, 2, -8, 9, -10, 5, -6, 7, -9, 8]
In[4]:=	DTCode[Knot[10, 145]]
Out[4]=	DTCode[4, 8, -12, -18, 2, -16, -20, -6, -10, -14]
In[5]:=	br = BR[Knot[10, 145]]
Out[5]=	BR[4, {-1, -1, -2, 1, -2, -1, -3, -2, 1, -2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 145]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 145]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 145]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 145]][t]
Out[10]=	-2 1 2 -3 + t + - + t + t t
In[11]:=	Conway[Knot[10, 145]][z]
Out[11]=	2 4 1 + 5 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 145]}
In[13]:=	{KnotDet[Knot[10, 145]], KnotSignature[Knot[10, 145]]}
Out[13]=	{3, -2}
In[14]:=	Jones[Knot[10, 145]][q]
Out[14]=	-10 -9 -8 -7 -2 -q + q - q + q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 145]}
In[16]:=	A2Invariant[Knot[10, 145]][q]
Out[16]=	-32 -30 -24 -14 -10 -8 -6 -q - q + q + q + q + q + q
In[17]:=	HOMFLYPT[Knot[10, 145]][a, z]
Out[17]=	4 6 8 10 4 2 8 2 4 4 2 a - a + a - a + 4 a z + a z + a z
In[18]:=	Kauffman[Knot[10, 145]][a, z]
Out[18]=	4 6 8 10 5 7 9 11 4 2 2 a + a + a + a - a z - 2 a z - 6 a z - 5 a z - 4 a z - 6 2 8 2 10 2 7 3 9 3 11 3 2 a z - 4 a z - 6 a z + 8 a z + 18 a z + 10 a z + 4 4 8 4 10 4 7 5 9 5 11 5 a z + 9 a z + 10 a z - 6 a z - 12 a z - 6 a z - 8 6 10 6 7 7 9 7 11 7 8 8 10 8 6 a z - 6 a z + a z + 2 a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 145]], Vassiliev[3][Knot[10, 145]]}
Out[19]=	{5, -12}
In[20]:=	Kh[Knot[10, 145]][q, t]
Out[20]=	-5 -3 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 21 9 17 8 17 7 15 6 13 6 15 5 q t q t q t q t q t q t 1 2 1 1 1 ------ + ------ + ------ + ----- + ----- 11 5 11 4 11 3 7 3 7 2 q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 145], 2][q]
Out[21]=	-29 -28 -27 2 -25 -24 2 2 -21 2 4 q - q - q + --- - q - q + --- - --- + q + --- - --- + 26 23 22 20 19 q q q q q 2 2 5 2 2 5 4 2 4 2 -8 --- + --- - --- + --- + --- - --- + --- + --- - --- + -- + q - 18 17 16 15 14 13 12 11 10 9 q q q q q q q q q q -7 -4 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

10_144 10_146