# 9 44

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 44's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 44 at Knotilus!

### Knot presentations

 Planar diagram presentation X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13 Gauss code -1, 4, -3, 1, -2, -9, 5, 3, -4, 2, 7, -8, 9, -5, 6, -7, 8, -6 Dowker-Thistlethwaite code 4 8 10 -14 2 -16 -6 -18 -12 Conway Notation [22,21,2-]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 9, width is 4,

Braid index is 4

[{10, 3}, {1, 7}, {8, 4}, {7, 10}, {6, 9}, {3, 8}, {2, 5}, {4, 6}, {5, 1}, {9, 2}]

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index ${\displaystyle \{4,5\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-3] Hyperbolic Volume 7.40677 A-Polynomial See Data:9 44/A-polynomial

[edit Notes for 9 44's three dimensional invariants] 9_44 has girth 4. See arXiv:math.GT/0508590 and a forthcoming paper by the same authors.

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 2}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{2}-4t+7-4t^{-1}+t^{-2}}$ Conway polynomial ${\displaystyle z^{4}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 17, 0 } Jones polynomial ${\displaystyle q^{2}-2q+3-3q^{-1}+3q^{-2}-2q^{-3}+2q^{-4}-q^{-5}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{2}a^{4}-a^{4}+z^{4}a^{2}+3z^{2}a^{2}+3a^{2}-2z^{2}-2+a^{-2}}$ Kauffman polynomial (db, data sources) ${\displaystyle a^{3}z^{7}+az^{7}+2a^{4}z^{6}+3a^{2}z^{6}+z^{6}+a^{5}z^{5}-2a^{3}z^{5}-3az^{5}-7a^{4}z^{4}-10a^{2}z^{4}-3z^{4}-3a^{5}z^{3}-a^{3}z^{3}+4az^{3}+2z^{3}a^{-1}+5a^{4}z^{2}+10a^{2}z^{2}+z^{2}a^{-2}+6z^{2}+a^{5}z+a^{3}z-az-za^{-1}-a^{4}-3a^{2}-a^{-2}-2}$ The A2 invariant ${\displaystyle -q^{16}+2q^{8}+q^{6}+q^{4}-1-q^{-4}+q^{-6}+q^{-8}}$ The G2 invariant ${\displaystyle q^{80}-q^{78}+2q^{76}-3q^{74}+q^{72}-4q^{68}+6q^{66}-5q^{64}+3q^{62}-q^{60}-4q^{58}+5q^{56}-4q^{54}+2q^{50}-4q^{48}+4q^{46}-4q^{42}+7q^{40}-5q^{38}+3q^{36}-2q^{32}+6q^{30}-4q^{28}+6q^{26}-2q^{24}+2q^{22}+4q^{20}-4q^{18}+4q^{16}-2q^{14}+q^{12}+3q^{10}-4q^{8}+2q^{6}-4q^{2}+5-6q^{-2}+2q^{-6}-6q^{-8}+5q^{-10}-3q^{-12}+q^{-14}+q^{-16}-3q^{-18}+2q^{-20}+q^{-24}+q^{-26}+q^{-32}+q^{-38}}$

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (0, -1)
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 ${\displaystyle 0}$ ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle {\frac {16}{3}}}$ ${\displaystyle {\frac {64}{3}}}$ ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 48}$ ${\displaystyle -{\frac {56}{3}}}$ ${\displaystyle {\frac {104}{3}}}$ ${\displaystyle -{\frac {16}{3}}}$ ${\displaystyle 0}$

V2,1 through V6,9 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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$0 is the signature of 9 44. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-1012χ
5       11
3      1 -1
1     21 1
-1    22  0
-3   11   0
-5  12    1
-7 11     0
-9 1      1
-111       -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$